# On the $abc$ Conjecture in Algebraic Number Fields

Andrew Scoones

###### Abstract

In this paper we prove a weak form of the $abc$ Conjecture generalised to
algebraic number fields. Given algebraic integers $a,\,b,\,c$ in a number
field $K$ satisfying $a+b=c$, we give an upper bound for the logarithm of the
projective height $H_{L}(a,\,b,\,c)$ in terms of norms of prime ideals
dividing $abc\mathcal{O}_{L}$, where $L$ is the Hilbert Class Field of $K$. In
many cases this allows us to give a bound in terms of the modified radical
$G:=G(a,\,b,\,c)$ as given by Masser in [21]. Furthermore, by employing a
recent result by Le Fourn, our estimates imply the upper bound

$\log H_{L}\left(a,\,b,\,c\right)<G^{\frac{1}{3}+\mathcal{C}\frac{\log\log\log
G}{\log\log G}},$

where $\mathcal{C}$ is an effectively computable constant. Further, given
conditions on the largest prime ideal dividing $abc\mathcal{O}_{L}$, we obtain
a sub-exponential bound for $H_{L}(a,\,b,\,c)$ in terms of the radical. As a
consequence of our results, we will give an application to the effective
Skolem-Mahler-Lech problem and give an improvement to a result by Lagarias and
Soundararajan on the XYZ Conjecture given in [16].

##  1 Introduction

Let $a,\,b,\,c:=a+b$ be positive, pairwise coprime integers and define the
radical

$G(a,\,b,\,c)=G=\prod_{\begin{subarray}{c}p\mid abc\\\ p\text{ a
prime}\end{subarray}}p.$

In 1988, based on a conjecture of Szpiro about elliptic curves and on the
Mason-Stothers Theorem in function fields [20], Osterlé conjectured that for
all positive integers as above, there exists a positive constant
$\mathcal{C}_{1}$ such that $c<G^{\mathcal{C}_{1}}$ [26]. Further, Masser
conjectured a stronger statement, that for all positive $\epsilon$ there
exists a constant $\mathcal{C}_{2}\left(\epsilon\right)$ such that
$c<\mathcal{C}_{2}\left(\epsilon\right)G^{1+\epsilon}$ [22]. While both of
these conjectures are referred to as the $abc$ conjecture, generally the
second form by Masser is focused on in the literature. These conjectures have
far-reaching implications across a range of topics; see [3] and the references
within. We also refer the reader to Chapter 14 of [2] and Chapter 5 of [36]
for a discussion of Vojta’s conjectures, a generalisation of the second
formulation of the $abc$ conjecture given above.

In [33], Stewart and Yu prove that there exists an effectively computable
positive constant $\mathcal{C}_{3}$ such that for all positive integers
$a,\,b,\,c=a+b$ with $(a,\,b,\,c)=1$ and $c>2$,

$\log c<G^{\frac{2}{3}+\frac{\mathcal{C}_{3}}{\log\log G}}.$

In [34], they were able to improve this result to

$\log{c}<\mathcal{C}_{4}G^{\frac{1}{3}}\left(\log G\right)^{3}.$

To do this required the use of Yu’s work extending lower bounds for linear
forms in logarithms to the $p$-adic setting [38][39]. We note Yu further
improved these bounds in a series of papers [40] [41][42][43]; indeed, we will
use results from [43], which make use of group varieties to strengthen the
relevant bounds.

Much work has also been done generalising the $abc$ conjecture to algebraic
number fields. Browkin discusses this direction of research in [4], while in
[21] Masser discusses some issues regarding adapting the radical $G$ to the
case of number fields. In [10], Györy shows that given a number field $K$ and
$a,\,b,\,c\in K^{*}$ with $a+b+c=0$ and any $\epsilon>0$, there is an
effectively computable $\mathcal{C}_{5}\left(\epsilon\right)$ such that

$\log\left(H_{K}\left(a,\,b,\,c\right)\right)<\mathcal{C}_{5}N_{K}\left(a,\,b,\,c\right)^{1+\epsilon},$

with

$H_{K}\left(a,\,b,\,c\right)=\prod_{\upsilon\in
M_{K}}\max\left(\left|a\right|_{\upsilon},\,\left|b\right|_{\upsilon},\,\left|c\right|_{\upsilon}\right),$

where $M_{K}$ is the set of normalised places of $K$ and

$N_{K}\left(a,\,b,\,c\right)=\prod_{\upsilon}\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}\right)^{\text{ord}_{\mathfrak{p}}\left(p\right)},$

where $p$ is a rational prime such that $\mathfrak{p}$ lies over $p$,
$\mathfrak{p}$ is the prime ideal of $\mathcal{O}_{K}$ corresponding to
$\upsilon\in M_{K}$ and $\upsilon$ is taken over all finite places such that
$\left|a\right|_{\upsilon},\,\left|b\right|_{\upsilon},\,\left|c\right|_{\upsilon}$
are not all equal. We note that this is the same as the modified support
(1.11) of [21]. Recall that the norm of a prime ideal $\mathfrak{p}$ of the
ring of integers $\mathcal{O}_{K}$ of a number field $K$ is defined to be

$\textrm{Nm}_{QQ}^{K}\left(\mathfrak{p}\right)=p^{f_{\mathfrak{p}}},$

where $f_{\mathfrak{p}}$ is the inertia degree of $\mathfrak{p}$ over $p$
[25]; that is

$f_{\mathfrak{p}}:=\left[\mathcal{O}_{K}/\mathfrak{p}:\mathbb{Z}/p\right].$

Initially we introduce some notation we will use throughout this article. Let
$K$ be a number field of degree $d$ and let
$a,\,b,\,c\in\mathcal{O}_{K}\backslash\left\\{0\right\\}$ be such that
$a+b+c=0$. Further, assume that $a\mathcal{O}_{K},\,b\mathcal{O}_{K}$ and
$c\mathcal{O}_{K}$ are pairwise coprime; that is
$a\mathcal{O}_{K}+b\mathcal{O}_{K}=\mathcal{O}_{K}$, and similarly for all
other pairs. Let $L=HCF(K)$ be the Hilbert Class Field of $K$ (that is, the
maximal abelian unramified extension of $K$ [7]) and let

$G=\prod_{\begin{subarray}{c}\mathfrak{P}\text{ prime ideal}\\\
\mathfrak{P}\subset\mathcal{O}_{L}\\\
\mathfrak{P}\mid\left(abc\right)\mathcal{O}_{L}\end{subarray}}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{P}\right).$

Let $\mathfrak{p}_{a}$ be the prime ideal of $\mathcal{O}_{L}$ of greatest
norm dividing $a\mathcal{O}_{L}$, and similarly for $\mathfrak{p}_{b}$ and
$\mathfrak{p}_{c}$. If $a$ is a unit, then we write that $\mathfrak{p}_{a}=1$
with norm 1, and similarly for $b$ and $c$. Write $\mathfrak{p}_{\max}$ for
the prime ideal of $\mathcal{O}_{L}$ of greatest norm dividing $G$. A priori,
this is equal to one of
$\mathfrak{p}_{a},\,\mathfrak{p}_{b},\,\mathfrak{p}_{c}$.

Let

$h(x)=\sum_{\upsilon\in M_{F}}\log^{+}\left|x\right|_{\upsilon}$

where $M_{F}$ is the set of places of the number field $F$ normalised so they
satisfy the product formula [2], and $\log^{+}(\alpha)=\max(\log\alpha,\,0)$.
Also, let

$H_{F}\left(x_{1},\,\dots,\,x_{n}\right)=\prod_{\upsilon\in
M_{F}}\max\left\\{\left|x_{1}\right|_{\upsilon},\,\dots,\,\left|x_{n}\right|_{\upsilon}\right\\}.$

It is worth pointing out that $H_{F}\left(x_{1},\dots,\,x_{n}\right)$ is the
projective height, so it gives the same value for any representative of
$(x_{1},\dots,\,x_{n})\in\mathbb{P}^{n-1}\left(F\right)$. Explicitly, this
means that for any $\left(a,\,b,\,c\right)\in\mathbb{P}^{2}\left(F\right)$ and
any $k\in F^{\times}$ we have that

$H_{F}\left(a,\,b,\,c\right)=H_{F}\left(ka,\,kb,\,kc\right).$ (1)

In particular, since in the set up of this article $c\neq 0$, we have that

$H_{L}\left(a,\,b,\,c\right)=H_{L}\left(\frac{a}{c},\,\frac{b}{c},\,1\right).$

We will generally be considering the height over the Hilbert Class Field $L$.
In this case, as $\left[K:\mathbb{Q}\right]=d$,

$h(x)=dh_{K}\log H_{L}(1,\,x),$

where $h_{K}$ is the class number of $K$. This follows as
$\left[L:K\right]=h_{K}$[2] [37]

We note that for any $x,\,y,\,z\in F$ where $F$ is an algebraic number field
of degree $d$,

$\displaystyle\log H_{F}(x,\,y,\,z)$ $\displaystyle=\log
H_{F}\left(\frac{x}{z},\,\frac{y}{z},\,1\right)$ $\displaystyle\leq
2d\max\left(h\left(\frac{x}{z}\right),\,h\left(\frac{y}{z}\right)\right).$ (2)

This follows directly from (4.3) of [10]. Furthermore, we will show in Section
3 that we can write $a=u_{a}a^{\prime}$ where $u_{a}$ is a unit such that

$\mathcal{C}_{6}\log\left|{\text{N}_{L/\mathbb{Q}}}\left(a^{\prime}\right)\right|\leq
h\left(a^{\prime}\right)\leq\mathcal{C}_{7}\log\left|{\text{N}_{L/\mathbb{Q}}}\left(a^{\prime}\right)\right|,$

where $\mathcal{C}_{6},\,\mathcal{C}_{7}$ are computable constants, and
similarly for $b$ and $c$. We assume without loss of generality that

$h(a^{\prime})\leq h(b^{\prime})\leq h(c^{\prime}).$ (3)

We initially prove the following main theorem.

######  Theorem 1.

Given the set up above, relabeling $a,\,b$ and $c$ if necessary to satisfy
(3), there exists an effectively computable constant $\mathcal{C}_{8}$
depending only on the field $K$ such that

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)^{\frac{1}{3}}G^{\mathcal{C}_{8}\frac{\log\log\log
G}{\log\log G}}.$ (4)

We will then give various corollaries to put the product of norms of prime
ideals in terms of the radical $G$, namely corollaries 1-7. Importantly, in
Corollary 7 we will give conditions that allow us to attain a sub-exponential
bound.

In later parts we will give related results that can be easier to manipulate
due to fewer prime ideals on the right hand side of the inequality, attaining
the following theorem.

######  Theorem 2.

Given the set up above, there exists an effectively computable number
$\mathcal{C}_{9}$ depending only on the field $K$ such that

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right)^{\frac{1}{2}}G^{\mathcal{C}_{9}\frac{\log\log\log
G}{\log\log G}}$
$\displaystyle=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)^{\frac{1}{2}}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)G^{\mathcal{C}_{9}\frac{\log\log\log
G}{\log\log G}}.$ (5)

We will then deduce corollaries 8-13, again giving conditions in Corollary 10
that give a sub-exponential bound in terms of the radical $G$.

We will then discuss how exploiting a method of Le Fourn [19] enables us to
reduce the dependency on prime ideals to give the following result with no
further conditions:

######  Theorem 3.

Given the set up above, there exists an effectively computable constant
$\mathcal{C}_{10}$ depending on $K$ such that

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{q}\right)\right)^{\frac{1}{3}}G^{\mathcal{C}_{10}\frac{\log\log\log
G}{\log\log G}},$ (6)

where $\mathfrak{p}_{c}^{\prime}$ is the prime ideal of third largest norm
dividing $c\mathcal{O}_{L}$ and $\mathfrak{q}$ is the prime ideal of
$\mathcal{O}_{L}$ of third largest norm dividing $bc\mathcal{O}_{L}$.

From here we will deduce that

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<G^{\frac{1}{3}+\mathcal{C}_{11}\frac{\log\log\log G}{\log\log
G}}.$ (7)

The results given in this paper, in particular Theorem 3, allow us to give a
new method of solving the effective Skolem-Mahler-Lech problem [28] of order
3. Additionally, we use Corollary 10 to expand on results by Lagarias and
Soundararajan regarding smooth solutions to the $abc$ equation [16]. We
briefly discuss both these problems here.

First we discuss the effective Skolem-Mahler-Lech problem. The problem is,
given a linear recurrence sequence, to decide whether said sequence contains
zeroes.

We recall that a linear recurrence sequence is a sequence $\left(a_{x}\right)$
of elements of a commutative ring with 1, $R$, satisfying a homogeneous linear
recurrence relation

$a_{x+n}=c_{1}a_{x+n-1}+\cdots+c_{n}a_{x},$

where $c_{1},\dots,\,c_{n}\in R$ [8]. We note, we will take $R$ to be an
algebraic number field.

The Skolem-Mahler-Lech Theorem is as follows.

######  Theorem 4 (Skolem-Mahler-Lech).

If a sequence of numbers satisfies a linear recurrence relation over a field
of characteristic zero, the the zeroes of this sequence can be decomposed into
the union of a finite set, and finitely many arithmetic progressions.

More comments on this theorem and a proof for sequences defined over the
rationals is given in [8], along with further references.

######  Remarks.

We note that there exists an algorithm to tell us if there are infinitely many
zeroes, and if so to find the decomposition of these zeros into periodic sets
guaranteed to exist by the Skolem–Mahler–Lech Theorem [1]. The effective
Skolem-Mahler-Lech problem then is to find an algorithm to determine whether
there are any non-periodic zeroes in a given linear recurrence sequence,
importantly in the case where these are the only zeroes [28]. This would allow
us to effectively answer whether a given linear recurrence relation contains
any zeroes.

Recall, given a linear recurrence relation

$a_{x+n}=c_{1}a_{x+n-1}+\cdots+c_{n}a_{x},$

and initial terms $a_{1},\dots,\,a_{n}\in R$, we can find a formula for the
$m$’th term. Given the recurrence relation, we find the characteristic
polynomial

$f(X)=X^{n}-c_{1}X^{n-1}-\cdots-c_{n-1}X-c_{n},$

with roots $r_{1},\dots,\,r_{l}$ with multiplicities $m_{1},\dots,\,m_{l}$
respectively. The $x$’th term of the sequence then is given by

$a_{x}=g_{1}(x)r_{1}^{x}+\cdots+g_{l}(x)r_{l}^{x},$

where $g_{i}(x)$ are polynomials with $\deg\left(g_{i}\right)\leq m_{i}-1$
which depend on the initial values $a_{1},\,\dots,\,a_{n}$.

In section 8 we will show how Theorem 3 gives a new method of determining if a
linear recurrence sequence of order 3 contains zeroes. For further references
of preexisting methods and results on this problem and related problems we
refer the reader to [14] [32] [27] and the references contained within them.

We now discuss the smooth $abc$ Conjecture, also referred to as the $xyz$
conjecture given by Lagarias and Soundararajan in [16]. Given a triple
$a,\,b,\,c:=a+b\in\mathbb{N}$, define the smoothness of the triple

$S(a,\,b,\,c):=\max\left\\{p\,:\,p\mid abc\right\\}.$

In [16], Lagarias and Soundararajan give the following conjecture, which they
refer to as the $xyz$ conjecture.

######  Conjecture 1 ($xyz$ conjecture).

There exists a positive constant $\kappa$ such that the following hold.

a) For each $\epsilon>0$ there are only finitely many integer solutions
$\left(X,\,Y,\,Z\right)$ to the equation $X+Y=Z$ with
$\left(X,\,Y,\,Z\right)=1$ and

$S\left(X,\,Y,\,Z\right)<\left(\log
H\left(X,\,Y,\,Z\right)\right)^{\kappa-\epsilon}.$

b) For each $\epsilon>0$ there are infinitely many integer solutions
$\left(X,\,Y,\,Z\right)$ to the equation $X+Y=Z$ with
$\left(X,\,Y,\,Z\right)=1$ and

$S\left(X,\,Y,\,Z\right)<\left(\log
H\left(X,\,Y,\,Z\right)\right)^{\kappa+\epsilon}.$

When a triple $\left(X,\,Y,\,Z\right)$ satisfies $X+Y=Z$ and
$\left(X,\,Y,\,Z\right)=1$, we will call the triple a primitive solution.
Lagarias and Soundararajan go on to conjecture that $\kappa=\frac{3}{2}$. They
note however that to prove the above conjecture, one need only prove that
there exists a $\kappa_{0}>0$ satisfying part a) and a $\kappa_{1}<\infty$
satisfying part b). As a) and b) are independent, monotonicity would then
imply the existence of a unique constant $\kappa$.

Lagarias and Soundararajan prove part b) assuming the Generalised Riemann
Hypothesis, and show that the $abc$ conjecture implies part a). Further, in
Corollary 1 of [15], Harper is able to show unconditionally that the
$xyz$-smoothness exponent $\kappa$ is finite, and showed that part b) of the
conjuecture holds.

Unconditionally, Lagarias and Soundararajan are able to give the following
result.

######  Theorem (Theorem 2.2 of [16]).

For each $\epsilon>0$ there are only finitely many solutions to $X+Y=Z$
satisfying $\left(X,\,Y,\,Z\right)=1$ and

$S\left(X,\,Y,\,Z\right)\leq\left(3-\epsilon\right)\log\log
H\left(X,\,Y,\,Z\right).$

The proof of this, and the improvement we will give below depend heavily on
Northcott’s Theorem. We recall that Northcott’s Theorem tells us that for any
given number field $K$ and $B\in\mathbb{R},\,B>0$, the set

$\left\\{\left(X,\,Y,\,Z\right)\in
K^{3}\,:\,H\left(X,\,Y,\,Z\right)<B\right\\}$

is finite [2].

Using results from this paper, we will improve this bound with the following
theorem.

######  Theorem 5.

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$\phi\left(x\right)<\log\log x$ with

$\lim_{x\rightarrow+\infty}\phi\left(x\right)=+\infty.$

Then there are finitely many solutions to $X+Y=Z$ satisfying
$\left(X,\,Y,\,Z\right)=1$ and

$S\left(X,\,Y,\,Z\right)\leq\log\log H\left(X,\,Y,\,Z\right)\frac{\log\log\log
H\left(X,\,Y,\,Z\right)}{\log\log\log\log
H\left(X,\,Y,\,Z\right)\phi\left(\log\log H\left(X,\,Y,\,Z\right)\right)}.$
(8)

We note that this result implies that there are only finitely many primitive
integer triples $\left(X,\,Y,\,Z\right)$ satisfying $X+Y=Z$ with

$S\left(X,\,Y,\,Z\right)<c\log\log H\left(X,\,Y,\,Z\right)$

for any constant $c\in\mathbb{R},\,c>0$. This is because for any such $c$,
there is a value $H$ such that for any $H\left(X,\,Y,\,Z\right)>H,$

$\frac{\log\log\log H\left(X,\,Y,\,Z\right)}{\log\log\log\log
H\left(X,\,Y,\,Z\right)\phi\left(\log\log H\left(X,\,Y,\,Z\right)\right)}>c,$

and by Northcott’s Theorem, there are only finitely many triples
$\left(X,\,Y,\,Z\right)$ satisfying $H\left(X,\,Y,\,Z\right)<H$. The statement
above then follows from Theorem 5, and along with the result given in Theorem
5 is also an improvement on the $3-\epsilon$ in Theorem 2.2 of [16].

######  Remarks.

Independently, using different methods, Györy has been able to show a similar
result to that of these results, but over the base field $K$ [12]. We will
discuss this further in section 7.

When this manuscript was completed, Professor Györy informed me about a
sharper $abc$ inequality over $\mathbb{Q}$ and imaginary quadratic number
fields by Mochizuki, Fesenko, Hoshi, Minamide and Porowski (submitted for
publication). However, this result relies on Mochizuki’s results in Inter-
universal Teichmüller Theory [23], the veracity of which is currently being
debated [31].

## Acknowledgements

I wish to thank Evgeniy Zorin for his advice and support on this paper, and to
thank Kálmán Györy for his helpful comments and encouragement on this paper. I
would also like to thank Sanju Velani for his insightful comments on the
paper.

##  2 Preliminary Lemmas

In this section we state pre-existing lemmas which we will repeatedly use
throughout the proof of Theorem 1. In the rest of the paper,
$\mathcal{C}_{11},\,\mathcal{C}_{12},\dots$ denote effectively computable
constants, and we will, where relevant, state what these constants depend on.
In many cases, the constants depend on properties determined by a certain
field; in these cases we will sometimes explicitly give which properties of
the field the constants depend on.

We note that by Lemma 1 of [6], we can find a system of fundamental units
$\eta_{1},\dots,\,\eta_{r}$ where $r$ is the unit rank of $F$ such that

$\prod_{i=1}^{r}h\left(\eta_{i}\right)\leq\mathcal{C}_{11}R_{F}$ (9)

where

$\mathcal{C}_{11}=\frac{\left(\left(r-1\right)!\right)^{2}}{2^{r-2}d^{r-1}}$

and $R_{F}$ is the regulator of the field $F$. Throughout this paper we will
use such a system of fundamental units, and often refer to them as ”the
fundamental units” of the field in question.

######  Lemma 1.

Let $F$ be a number field of degree $d$, and let
$\alpha\in\mathcal{O}_{F}\backslash\mathcal{O}_{F}^{*}$. Then there is an
effectively computable number $\mathcal{C}_{12}\left(F\right)$, depending on
the fundamental units of $\mathcal{O}_{F}$, and an
$\epsilon\in\mathcal{O}_{F}^{*}$ such that

${\vrule height=5.70554pt,width=0.4pt,depth=0.0pt\vrule
height=5.70554pt,width=12.45605pt,depth=-5.30554pt\hbox
to0.0pt{\hss$\epsilon\alpha$\kern 1.0pt}\vrule
height=5.70554pt,width=0.4pt,depth=0.0pt}\leq\mathcal{C}_{12}\left|{\text{N}_{F/\mathbb{Q}}}\left(\alpha\right)\right|^{1/d}$

where  $\alpha$ denotes the house of $\alpha$.

We recall that  $\alpha$ , the house of $\alpha$, is defined to be the maximal
absolute value of the conjugates of $\alpha$ over $\mathbb{C}$.

###### Proof.

This is Lemma 1.3.8 from [24]. ∎

######  Lemma 2.

Let $F$ be an algebraic number field of degree $d$ with set of normalised
places $M_{F}$, and let $S$ be a finite subset of $M_{F}$ which contains
$S_{\infty}$, the set of infinite places. Let $s$ be the cardinality of $S$,
$\mathfrak{p}_{1},\,\dots,\,\mathfrak{p}_{t}$ the prime ideals corresponding
to finite places of $S$ and let
$P=\max_{i}\textrm{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{i}\right)$. Further
let $R_{S}$ be the $S$-regulator of $F$. We note that
$R_{S}=i_{S}R\prod_{i=1}^{t}\log\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{i}\right)$,
where $i_{S}$ is a positive divisor of the class number $h_{F}$ of $F$ and $R$
is the regulator of $F$ (cf. [10][9][13]). Given $\alpha,\,\beta$, non-zero
elements of $F$, we consider the $S$-unit equation $\alpha x+\beta y=1$ in
$x,\,y$, where $x,\,y$ are $S$-units. Let $r$ denote the unit rank of $F$ and
let $\mathcal{R}=\max\left\\{h_{F},\,\mathcal{C}_{13}(r,\,d)R\right\\}$, where
$\mathcal{C}_{13}\left(r,\,d\right)$ is given explicitly in [10]. Then, if
$t=0$, all solutions $x,\,y$ of the above equation satisfy

$\max\left\\{h(x),\,h(y)\right\\}\leq\mathcal{C}_{13}\left(r,\,d\right)\max\left\\{h(\alpha),\,h(\beta),\,1\right\\}.$

If $t>0$ then we obtain

$\max\left\\{h(x),\,h(y)\right\\}\leq\mathcal{C}_{14}\left(r,\,d\right)h_{F}R\left(\log^{*}R\right)\mathcal{R}^{t+1}\left(\log^{*}\mathcal{R}\right)\left(\frac{P}{\log^{*}P}\right)R_{S}\max\left\\{h(\alpha),\,h(\beta),\,1\right\\},$

where $\mathcal{C}_{14}\left(r,\,d\right)$ is also explicitly given in [10].

###### Proof.

This is part of Theorem A from [10]. ∎

######  Lemma 3.

Let $\alpha_{1},\,\dots,\,\alpha_{n}$ be algebraic numbers and $F$ a number
field of degree $d$ containing $\alpha_{1},\,\dots,\,\alpha_{n}$. Let
$\mathfrak{p}$ be a prime ideal of $\mathcal{O}_{F}$ lying above the rational
prime $p$ with ramification index $e_{\mathfrak{p}}$ and residue class degree
$f_{\mathfrak{p}}$. For $\alpha\in F,\,\alpha\neq 0$, we write
$\text{ord}_{\mathfrak{p}}\left(\alpha\right)$ for the exponent to which
$\mathfrak{p}$ divides the principal fractional ideal generated by $\alpha$ in
$F$, and we set $\text{ord}_{\mathfrak{p}}\left(0\right)=+\infty$. Let
$b_{1},\,\dots b_{n}$ be integers, and set
$\Theta=\alpha_{1}^{b_{1}}\cdots\alpha_{n}^{b_{n}}-1$. Assume that $\Theta\neq
0$. Finally, set
$h^{\prime}\left(\alpha_{j}\right)=\max\left\\{h\left(\alpha_{j}\right),\,\frac{1}{16e^{2}d^{2}}\right\\}$.
Then

$\text{ord}_{\mathfrak{p}}\left(\Theta\right)<\left(16ed\right)^{2\left(n+1\right)}n^{5/2}\log\left(2nd\right)\log\left(2d\right)\cdot
e_{\mathfrak{p}}^{n}\frac{\emph{\text{Nm}}_{\mathbb{Q}}^{F}\left(\mathfrak{p}\right)}{\left(\log\emph{\text{Nm}}_{\mathbb{Q}}^{F}\left(\mathfrak{p}\right)\right)^{2}}\prod_{i=1}^{n}h^{\prime}\left(\alpha_{i}\right)\log
B,$

where
$B=\max\left\\{\left|b_{1}\right|,\,\dots,\,\left|b_{n}\right|,\,3\right\\}$.

###### Proof.

This is a consequence of the main theorem of [43], given on page 190. ∎

######  Lemma 4.

Let $F$ be a number field with ring of integers $\mathcal{O}_{F}$. Apply a
total ordering to the prime ideals, so that if
$\emph{\text{Nm}}_{\mathbb{Q}}^{F}~{}\left(\mathfrak{x}\right)>\emph{\text{Nm}}_{\mathbb{Q}}^{F}\left(\mathfrak{y}\right)$,
then $\mathfrak{x}\succ\mathfrak{y}$. Arbitrarily order ideals of the same
norm. Then there is an effectively computable positive constant
$\mathcal{C}_{15}$ such that for every positive integer $r$ we have

$\prod_{i=1}^{r}\frac{\emph{\text{Nm}}_{\mathbb{Q}}^{F}~{}\left(\mathfrak{p}_{i}\right)}{\log\emph{\text{Nm}}_{\mathbb{Q}}^{F}~{}\left(\mathfrak{p}_{i}\right)}>\left(\frac{r}{\mathcal{C}_{15}}\right)^{r}.$

###### Proof.

Let $\pi_{F}(x)$ denote the number of prime ideals in number field $F$ of norm
less than or equal to $x$. By the Landau Prime Ideal Theorem [17], we know
that $\pi_{F}(x)\sim\frac{x}{\log x}$. Partially order the prime ideals as in
the statement of the Lemma. Then by Landau,

$\pi_{F}\left(\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)\right)\sim\frac{\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)}{\log\left(\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)\right)}.$

Thus by Landau, there exists an effectively computable number
$\mathcal{C}_{16}$ such that

$\frac{\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)}{\log\left(\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)\right)}>j/\mathcal{C}_{16}.$

Thus, using the inequality $r!\geq\left(r/e\right)^{r}$, we see that

$\prod_{j=1}^{r}\frac{\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)}{\log\left(\text{Nm}_{\mathbb{Q}}^{F}\left(\mathfrak{p}_{j}\right)\right)}>\frac{r!}{\mathcal{C}_{16}^{r}}\geq\left(\frac{r}{\mathcal{C}_{17}e}\right)^{r}\geq\left(\frac{r}{\mathcal{C}_{18}}\right)^{r},$

which proves the claim in the lemma. ∎

We finally state a lemma that we will use to tidy our end arguments.

######  Lemma 5.

If $x,\,a\in\mathbb{R}$ with $\frac{a}{\log a}<x$, then
$a<\max\left\\{e,\,2x\log x\right\\}$.

###### Proof.

If $\frac{a}{\log a}<x$, then

$a<x\log a.$ (10)

As $\frac{a}{\log a}<x$, we see that $\log a-\log\log a<\log x$. Further, we
can show that $\frac{\log a}{2}<\log a-\log\log a$.

Combining these we get that

$\log a<2\log x.$ (11)

Multiplying together (10) and (11) we obtain that $a<2x\log x$. ∎

##  3 Proof of the Main Theorem

Let $K$ be a number field with ring of integers $\mathcal{O}_{K}$, class
number $h_{K}$, and Hilbert Class Field $L$. Let
$\left[K:\mathbb{Q}\right]=d$, so by the tower law, $[L:\mathbb{Q}]=h_{K}d$.

Take $a,\,b,\,c:=-a-b\in\mathcal{O}_{K}$ so that

$a+b+c=0,$ (12)

with the assumption that $a\mathcal{O}_{K},\,b\mathcal{O}_{K}$ and
$c\mathcal{O}_{K}$ are coprime. We write

$\displaystyle a\mathcal{O}_{K}$
$\displaystyle=\mathfrak{p}_{1}^{e_{1}}\cdots\mathfrak{p}_{t}^{e_{t}}$
$\displaystyle b\mathcal{O}_{K}$
$\displaystyle=\mathfrak{q}_{1}^{f_{1}}\cdots\mathfrak{q}_{u}^{f_{u}}$
$\displaystyle c\mathcal{O}_{K}$
$\displaystyle=\mathfrak{r}_{1}^{g_{1}}\cdots\mathfrak{r}_{v}^{g_{v}},$ (13)

where $\mathfrak{p}_{i},\,\mathfrak{q}_{j},\,$ and $\mathfrak{r}_{k}$ are
prime ideals of $\mathcal{O}_{K}$ and $e_{i},\,f_{j},\,g_{k}$ are integers.

A key property of $L=HCF(K)$ is that every ideal $\mathfrak{I}$ of
$\mathcal{O}_{K}$ is principal in $\mathcal{O}_{L}$; that is
$\mathfrak{I}\mathcal{O}_{L}=\alpha\mathcal{O}_{L}$ for some
$\alpha\in\mathcal{O}_{L}$. By Lemma 1, we can pick the generator of each
ideal so that if $\alpha$ is the generator, then

$h\left(\alpha\right)\leq\log{\vrule
height=5.70554pt,width=0.4pt,depth=0.0pt\vrule
height=5.70554pt,width=8.39702pt,depth=-5.30554pt\hbox
to0.0pt{\hss$\alpha$\kern 1.0pt}\vrule
height=5.70554pt,width=0.4pt,depth=0.0pt}\leq\log\left(\mathcal{C}_{19}\left(K\right)\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha\right)\right|^{1/d}\right)=\mathcal{C}_{20}\left(K\right)\log\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha\right)\right|.$
(14)

We note that the dependence of the constants is on $K$ rather than $L$, as $L$
is uniquely determined by $K$. Further, for such algebraic $\alpha$ we have

$\displaystyle\log\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha\right)\right|\leq
dh\left(\alpha\right),$

giving us that

$\mathcal{C}_{21}\left(K\right)\log\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha\right)\right|\leq
h\left(\alpha\right)\leq\mathcal{C}_{22}\left(K\right)\log\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha\right)\right|.$
(15)

Recalling this, we write

$\displaystyle\mathfrak{p}_{i}\mathcal{O}_{L}$
$\displaystyle=a_{i}\mathcal{O}_{L}$
$\displaystyle\mathfrak{q}_{j}\mathcal{O}_{L}$
$\displaystyle=b_{j}\mathcal{O}_{L}$
$\displaystyle\mathfrak{r}_{k}\mathcal{O}_{L}$
$\displaystyle=c_{k}\mathcal{O}_{L},$ (16)

where $a_{i},\,b_{j},\,c_{k}$ satisfy (14).

We can also write $a\mathcal{O}_{L},\,b\mathcal{O}_{L}$ and $c\mathcal{O}_{L}$
as a product of prime ideals of $\mathcal{O}_{L}$. Knowing this, we will write

$G=\prod_{\begin{subarray}{c}\mathfrak{P}\text{ prime ideal}\\\
\mathfrak{P}\subset\mathcal{O}_{L}\\\
\mathfrak{P}\mid\left(abc\right)\mathcal{O}_{L}\end{subarray}}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{P}\right).$
(17)

Note that by our assumptions this is equivalent to taking the field to be $L$
in equation (1.7) of [21]. Further we will denote the prime ideal of
$\mathcal{O}_{L}$ of largest norm dividing $a\mathcal{O}_{L}$ by
$\mathfrak{p}_{a}$, and similarly for $b$ and $c$.

From (3) we can write $a,\,b,\,c$ as follows:

$\displaystyle a$ $\displaystyle=u_{a}a_{1}^{e_{1}}\cdots a_{t}^{e_{t}}$
$\displaystyle b$ $\displaystyle=u_{b}b_{1}^{f_{1}}\cdots b_{u}^{f_{u}}$
$\displaystyle c$ $\displaystyle=u_{c}c_{1}^{g_{1}}\cdots c_{v}^{g_{v}}$ (18)

where $u_{a},\,u_{b}$ and $u_{c}$ are units of $\mathcal{O}_{L}$. We will also
often write $u_{a}a^{\prime}+u_{b}b^{\prime}+u_{c}c^{\prime}=0$ where
$a^{\prime}=\prod_{i=1}^{s}a_{i}$ and similarly for $b^{\prime}$ and
$c^{\prime}$. After relabeling if needed, we can assume that (3) holds. We
note that if $h\left(c^{\prime}\right)\leq 1$ then straightforwardly the claim
holds. This is because necessarily $h(a^{\prime})\leq h(b^{\prime})\leq
h(c^{\prime})\leq 1$ and Theorem 1 readily follows from (1). Similarly, after
some work we do below finding a bound on $h(c^{\prime})$, we see that if
$h(b^{\prime})\leq 1$ then again we’ll find the claim straightforwardly
follows, and the same logic will hold if $h(a^{\prime})\leq 1$. Thus we assume
in the following that

$1<h(a^{\prime})\leq h(b^{\prime})\leq h(c^{\prime}).$ (19)

Dividing through by $u_{c}c^{\prime}=c$ in (12) we obtain that

$-\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}-\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}=1,$
(20)

so we are in a position to apply Lemma 2.

Before doing this, we note that by the remarks around (1), $\log
H_{L}\left(a,\,b,\,c\right)=\log
H_{L}\left(\frac{a}{c},\,\frac{b}{c},\,1\right)$. This allows us to move
between representatives of the projective point
$\left[a:\,b:\,c\right]\in\mathbb{P}^{2}\left(L\right)$.

First, following the notation from Lemma 2, initially take $S$ to be the set
of infinite places of $L$. Applying this to (20), we obtain that

$\displaystyle\max\left\\{h\left(-\frac{u_{a}}{u_{c}}\right),\,h\left(-\frac{u_{b}}{u_{c}}\right)\right\\}$
$\displaystyle=\max\left\\{h\left(\frac{u_{a}}{u_{c}}\right),\,h\left(\frac{u_{b}}{u_{c}}\right)\right\\}$
$\displaystyle\leq\mathcal{C}_{23}\left(K\right)\max\left\\{h\left(\frac{a^{\prime}}{c^{\prime}}\right),\,h\left(\frac{b^{\prime}}{c^{\prime}}\right),\,1\right\\}.$
(21)

We note similar bounds also hold if we divide (12) through by
$u_{a}a^{\prime}$ or $u_{b}b^{\prime}$. Recall that $h\left(xy\right)\leq
h\left(x\right)+h\left(y\right)$ [2][37]. By this fact and our assumptions on
$h(a^{\prime}),\,h(b^{\prime})\text{ and }h(c^{\prime})$, we deduce that

$\max\left\\{h\left(\frac{a^{\prime}}{c^{\prime}}\right),\,h\left(\frac{b^{\prime}}{c^{\prime}}\right),\,1\right\\}\leq\max\left\\{h\left(a^{\prime}\right)+h\left(c^{\prime}\right),\,h\left(b^{\prime}\right)+h\left(c^{\prime}\right)\right\\}\leq
2h\left(c^{\prime}\right).$ (22)

It thus follows that

$\displaystyle h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right)$
$\displaystyle\leq
h\left(\frac{u_{a}}{u_{c}}\right)+h\left(a^{\prime}\right)+h\left(c^{\prime}\right)$
$\displaystyle\leq\mathcal{C}_{24}h\left(c^{\prime}\right)+2h\left(c^{\prime}\right)$
$\displaystyle=\mathcal{C}_{25}h\left(c^{\prime}\right).$ (23)

Similarly, we also obtain that

$\displaystyle h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)$
$\displaystyle\leq\mathcal{C}_{26}h\left(c^{\prime}\right).$ (24)

We again apply Lemma 2, but this time with a different set of $S$-units. Now
let $S=S_{\infty}\cup\left\\{\mathfrak{p}\,:\,\mathfrak{p}\mid
c\mathcal{O}_{L}\right\\}$ where $\mathfrak{p}$ refers both to the prime ideal
$\mathfrak{p}$ and its corresponding place. Applying Lemma 2 in this case to
(20), we obtain that

$\displaystyle\max\left\\{h\left(\frac{u_{a}}{u_{c}c^{\prime}}\right),\,h\left(\frac{u_{b}}{u_{c}c^{\prime}}\right)\right\\}$
$\displaystyle\leq\mathcal{C}_{27}\left(K\right)\left(\frac{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)}{\log^{*}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
c\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)\max\left\\{h\left(a^{\prime}\right),\,h\left(b^{\prime}\right),\,1\right\\}$
$\displaystyle\leq\mathcal{C}_{27}\left(K\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h\left(b^{\prime}\right).$
(25)

Again, $L$ is defined by $K$ uniquely, hence the dependence of the constants
here on $K$ rather than $L$. Further, by (19), we have that
$\max\left\\{h\left(a^{\prime}\right),\,h\left(b^{\prime}\right),\,1\right\\}=h\left(b^{\prime}\right)$.

From (3) we obtain that

$\displaystyle h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right)$
$\displaystyle\leq
h\left(\frac{u_{a}}{u_{c}c^{\prime}}\right)+h\left(a^{\prime}\right)$
$\displaystyle\leq\mathcal{C}_{27}\left(K\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h\left(b^{\prime}\right)+h\left(a^{\prime}\right)$
$\displaystyle\leq\left(\mathcal{C}_{27}\left(K\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)+1\right)h\left(b^{\prime}\right)$
$\displaystyle\leq\mathcal{C}_{28}\left(K\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(b^{\prime}).$
(26)

Similarly, we find that

$\displaystyle h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)$
$\displaystyle\leq\mathcal{C}_{29}\left(K\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(b^{\prime}).$
(27)

We now choose another set $S$, this time containing the infinite places and
the finite places corresponding to the prime ideals dividing
$bc\mathcal{O}_{L}$; that is,
$S=S_{\infty}\cup\left\\{\mathfrak{p}\,:\,\mathfrak{p}\mid
bc\mathcal{O}_{L}\right\\}$. Applying Lemma 2 to (20) with this $S$ we obtain
that

$\displaystyle\max\left\\{h\left(\frac{u_{a}}{u_{c}c^{\prime}}\right),\,h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)\right\\}$
$\displaystyle\leq\mathcal{C}_{30}\left(K\right)\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid bc\mathcal{O}_{L}\\\
\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)\max\left\\{h\left(a^{\prime}\right),\,1\right\\}.$
(28)

From (19) and (28) we obtain that

$\displaystyle h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right)$
$\displaystyle\leq
h\left(\frac{u_{a}}{u_{c}c^{\prime}}\right)+h\left(a^{\prime}\right)$
$\displaystyle\leq\mathcal{C}_{30}\left(K\right)\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid bc\mathcal{O}_{L}\\\
\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h\left(a^{\prime}\right)+h\left(a^{\prime}\right)$
$\displaystyle\leq\left(\mathcal{C}_{30}\left(K\right)\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid bc\mathcal{O}_{L}\\\
\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)+1\right)h\left(a^{\prime}\right)$
$\displaystyle\leq\left(\mathcal{C}_{31}\left(K\right)\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid bc\mathcal{O}_{L}\\\
\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)\right)h\left(a^{\prime}\right).$
(29)

Similarly, we find that

$\displaystyle h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)$
$\displaystyle\leq\mathcal{C}_{32}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(a^{\prime}).$
(30)

By consideration of (3), (24), we see that

$\displaystyle\max\left\\{h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right),\,h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)\right\\}$
$\displaystyle<\max\left\\{\mathcal{C}_{25},\,\mathcal{C}_{26}\right\\}h\left(c^{\prime}\right)$
$\displaystyle=\mathcal{C}_{33}h\left(c^{\prime}\right),$ (31)

while (3) and (27) show that

$\displaystyle\max\left\\{h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right),\,h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)\right\\}$
$\displaystyle<\max\left\\{\mathcal{C}_{28},\mathcal{C}_{29}\right\\}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(b^{\prime})$
$\displaystyle=\mathcal{C}_{34}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(b^{\prime}).$
(32)

In the same way, it follows from (3) and (30) that

$\displaystyle\max\left\\{h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right),\,h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)\right\\}$
$\displaystyle<\max\left\\{\mathcal{C}_{31},\mathcal{C}_{32}\right\\}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(a^{\prime})$
$\displaystyle=\mathcal{C}_{35}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)h(a^{\prime}).$
(33)

We next prove the following lemma.

######  Lemma 6.

Let $\alpha\in\left\\{a,\,b,\,c\right\\}$. Then

$h\left(\alpha^{\prime}\right)\leq\mathcal{C}_{36}\left(\max_{\mathfrak{p}\mid\left<\alpha\right>_{L}}\text{ord}_{\mathfrak{p}}\left(\alpha\right)\right)\log
G.$ (34)

###### Proof.

If $\alpha$ is a unit, then we define
$\max\,\text{ord}_{\mathfrak{p}}\left(\alpha\right):=1$.

By construction of $\alpha^{\prime}$, we have that
$h\left(\alpha^{\prime}\right)\leq\mathcal{C}_{37}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\alpha\mathcal{O}_{L}\right)$.
This follows from the fact that

$\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha^{\prime}\right)\right|=\left|{\text{N}_{L/\mathbb{Q}}}\left(\alpha\right)\right|=\text{Nm}_{\mathbb{Q}}^{L}\left(\alpha\mathcal{O}_{L}\right).$

We write a factorisation of
$\alpha\mathcal{O}_{L}=\mathfrak{P}_{1,L}^{g_{1,L}}\cdots\mathfrak{P}_{u_{L},L}^{g_{u_{L},L}}$
into prime ideals of $\mathcal{O}_{L}$. Note this may be different to the
ideals given in (3), as the ideals in (3) may not be prime. Working with this
prime factorisation, we obtain that

$\displaystyle\log\text{Nm}_{\mathbb{Q}}^{L}\left(\alpha\mathcal{O}_{L}\right)$
$\displaystyle=\log\left(\prod_{i=1}^{u_{L}}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{i,L}^{u_{i},L}\right)\right)$
$\displaystyle=\sum_{i=1}^{u_{L}}u_{i,L}\log\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{i,L}\right)\right)$
$\displaystyle\leq\left(\max_{\mathfrak{p}\mid\left<\alpha\right>_{L}}\text{ord}_{\mathfrak{p}}\left(\alpha\right)\right)\log
G.$

The claim then follows. ∎

It follows immediately from (1), (3) and Lemma 6 that

$\frac{\log H_{L}\left(a,\,b,\,c\right)}{\mathcal{C}_{33}\log
G}\leq\max_{\mathfrak{p}\mid\left<c\right>_{L}}\text{ord}_{\mathfrak{p}}\left(c\right).$
(35)

Similarly, it follows from (1), (3) and Lemma 6 that

$\frac{\log H_{L}\left(a,\,b,\,c\right)}{\mathcal{C}_{34}\log
G\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)}\leq\max_{\mathfrak{p}\mid\left<b\right>_{L}}\text{ord}_{\mathfrak{p}}\left(b\right).$
(36)

Further, it follows from (1), (3) and Lemma 6 that

$\frac{\log H_{L}\left(a,\,b,\,c\right)}{\mathcal{C}_{35}\log
G\,\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\left(\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)}\leq\max_{\mathfrak{p}\mid\left<b\right>_{L}}\text{ord}_{\mathfrak{p}}\left(a\right).$
(37)

We will use Lemma 3 to establish upper bounds for the right-hand sides of
(35), (36) and (37). In order to do this we need to write
$\text{ord}_{\mathfrak{p}}\left(c\right)$,
$\text{ord}_{\mathfrak{p}}\left(b\right)$ and
$\text{ord}_{\mathfrak{p}}\left(a\right)$ in a form where we’re able to use
Lemma 3.

By the coprimeness of $a\mathcal{O}_{L},\,b\mathcal{O}_{L}$ and
$c\mathcal{O}_{L}$ we see that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(c\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\frac{c}{b}\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\frac{-a-b}{b}\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{a}{b}-1\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{u_{a}}{u_{b}}a_{1}^{e_{1}}\cdots
a_{t}^{e_{t}}b_{1}^{f_{1}}\dots b_{u}^{f_{u}}-1\right).$ (38)

Similarly,

$\displaystyle\text{ord}_{\mathfrak{p}}\left(b\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\frac{b}{a}\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\frac{-a-c}{a}\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{c}{a}-1\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{u_{c}}{u_{a}}c_{1}^{g_{1}}\cdots
c_{v}^{g_{v}}a_{1}^{-e_{1}}\dots a_{t}^{-e_{t}}-1\right),$ (39)

and

$\displaystyle\text{ord}_{\mathfrak{p}}\left(a\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\frac{a}{c}\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\frac{-b-c}{c}\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{b}{c}-1\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{u_{b}}{u_{c}}b_{1}^{f_{1}}\cdots
b_{u}^{f_{u}}c_{1}^{-g_{1}}\dots c_{v}^{-g_{v}}-1\right).$ (40)

To apply Lemma 3 we need to bound exponents $e_{i},\,f_{j},\,g_{k}$ and bound
the heights of the units $h\left(u_{a}\right),\,h\left(u_{b}\right)$ and
$h\left(u_{c}\right)$.

First note that

$\displaystyle\max\left\\{\text{ord}_{\mathfrak{p}}\left(a\right),\,\text{ord}_{\mathfrak{p}}\left(b\right),\,\text{ord}_{\mathfrak{p}}\left(c\right)\right\\}\leq\log
H_{L}\left(a,\,b,\,c\right),$ (41)

directly from the definition of $H_{L}$.

This follows from the definitions of projective and absolute logarithmic
heights.

In order to use Yu’s bound, we need to manage the heights of $u_{a},\,u_{b}$
and $u_{c}$. To do this, we will use fundamental units of
$\mathcal{O}_{L}^{*}$. By Dirichlet’s Unit Theorem, there exist fundamental
units $\xi_{1},\dots,\xi_{r}$ of $\mathcal{O}_{L}$, where $r$ is the unit rank
of $\mathcal{O}_{L}$ such that all units $u$ of $\mathcal{O}_{L}$ can be
written $u=\mu\xi_{1}^{\delta_{1}}\cdots\xi_{r}^{\delta_{r}}$, with $\mu$ a
root of unity. We note again that finding a set of fundamental units is
computable, for example see [5], and we can find a nice system satisfying (9).
Indeed, we could use any system of fundamental units, but this choice is
helpful should one wish to explicitly compute the constants. Thus once found,
the product $\prod_{i=1}^{r}h^{\prime}(\xi_{i})$ we will obtain applying Lemma
3 can be upper bounded constants depending on the field. It remains to find an
upper bound for $\max_{i}\delta_{i}$.

Note that (3) gives us that

$h\left(\frac{u_{a}}{u_{c}}\right)\leq\mathcal{C}_{38}h\left(c^{\prime}\right).$

Further,

$h\left(c^{\prime}\right)\leq\mathcal{C}_{39}\left(K\right)\log\left|{\text{N}_{L/\mathbb{Q}}}\left(c\right)\right|\leq\mathcal{C}_{40}\left(K\right)h\left(c\right)\leq\mathcal{C}_{41}\left(K\right)\log
H_{L}(a,\,b,\,c).$

It follows from the above comments that

$h\left(\frac{u_{a}}{u_{c}}\right)\leq\mathcal{C}_{42}\left(K\right)\log
H_{L}(a,\,b,\,c).$ (42)

Let $L$ have $r_{1}$ real embeddings $\epsilon_{1},\dots,\,\epsilon_{r_{1}}$
and $2r_{2}$ complex embeddings
$\epsilon_{r_{1}+1},\,\overline{\epsilon_{r_{1}+1}},\dots,\epsilon_{r_{2}},\,\overline{\epsilon_{r_{2}}}$.
By Dirichlet’s Unit Theorem, there are $r:=r_{1}+r_{2}-1$ fundamental units
$\xi_{1},\dots,\xi_{r}$ such that for any unit $u\in\mathcal{O}_{L}^{*}$,
$u=\mu\xi_{1}^{\delta_{1}}\cdots\xi_{r}^{\delta_{r}}$ where $\mu$ is a root of
unity in $L$ and $\delta_{i}\in\mathbb{Z}$ for all $i$.

We now prove a lemma that gives an upper bound for
$\max_{i}\left|\delta_{i}\right|$.

######  Lemma 7.

Given the set up above,

$\max_{i}\left|\delta_{i}\right|\leq\mathcal{C}_{43}\left(K\right)\log
H_{L}(a,\,b,\,c).$ (43)

###### Proof.

Recall that there are $r+1$ distinct embeddings of $L$ into $\mathbb{C}$. Let
us denote these embeddings by $e_{i}$, $i=1\dots,r+1$ . Furthermore for all
$i=1\dots,r+1$, let us define

$\displaystyle\varepsilon_{i}:L$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle\alpha$ $\displaystyle\to\log|e_{i}(\alpha)|.$

Let $u$ be an element of $L$. We see that

$h\left(u\right)=\frac{1}{2}\sum_{i=1}^{r+1}N_{i}\left|\varepsilon_{i}\left(u\right)\right|,$

where $N_{i}=1$ if the image of $e_{i}$ is a subset of $\mathbb{R}$, and
$N_{i}=2$ in the complimentary case, when the image of $e_{i}$ is not a subset
of $\mathbb{R}$. This is because

$\displaystyle
h(u)=\sum_{i=1}^{r+1}N_{i}\log\max\left(\left|e_{i}(u),\,1\right|\right)=-\sum_{i=1}^{r+1}N_{i}\log\min\left(\left|e_{i}(u),\,1\right|\right),$

where $N_{i}$ is defined as above. Thus

$\displaystyle 2h(u)$
$\displaystyle=\sum_{i=1}^{r+1}N_{i}\log\max\left(\left|e_{i}(u),\,1\right|\right)-\sum_{i=1}^{r+1}N_{i}\log\min\left(\left|e_{i}(u),\,1\right|\right)$
$\displaystyle=\sum_{i=1}^{r+1}N_{i}\left|\log\left|e_{i}(u)\right|\right|$
$\displaystyle=\sum_{i=1}^{r+1}N_{i}\left|\varepsilon_{i}(u)\right|,$

and thus the identity follows.

We now take advantage of some properties of $u_{a},\,u_{b}$ and $u_{c}$ so we
will write them explicitly. In what follows we use the case
$u=\frac{u_{a}}{u_{c}}$, but it is true for the other relevant quotients of
$u_{a},\,u_{b}$ and $u_{c}$. Write
$\frac{u_{a}}{u_{c}}=\mu\xi_{1}^{\delta_{1}}\cdots\xi_{r}^{\delta_{r}}$ where
$\mu$ is a root of unity. From the comments above it follows that

$\displaystyle h\left(\frac{u_{a}}{u_{c}}\right)$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{r+1}N_{i}\left|\varepsilon_{i}\left(\mu\prod_{j=1}^{r}\xi_{j}^{\delta_{j}}\right)\right|$
(44)
$\displaystyle=\frac{1}{2}\sum_{i=1}^{r+1}N_{i}\left|\sum_{j=1}^{r}\delta_{j}\varepsilon_{i}(\xi_{j})\right|,$

where we lose the $\mu$ as it is a root of unity, so for all $i$,
$\varepsilon_{i}\left(\mu\right)=0$.

From (42), we know that
$h\left(\frac{u_{a}}{u_{c}}\right)\leq\mathcal{C}_{42}\log H_{L}(a,\,b,\,c)$,
giving us an upper bound for the absolute logarithmic height of the unit.
Along with (44), this implies that, for all $i=1,\dots,r+1$, we have

$\left|\sum_{j=1}^{r}\delta_{j}\varepsilon_{i}(\xi_{j})\right|<\mathcal{C}_{44}\left(K\right)\log
H_{L}(a,\,b,\,c).$ (45)

That is, (42) implies that all the exponents $\delta_{j}$, $j=1,\dots,r$
satisfy (45). This holds for all $\varepsilon_{i},\,i=1,\dots,\,r+1$, so (45)
gives us a system of $r+1$ inequalities.

We pick any $r$ inequalities of $r+1$ in the system (45). For the sake of
concreteness, let us take the first $r$ inequalities. We are going to deduce
the upper bound for the system of inequalities (45) where $i=1,\dots,r$. Note
that the left-hand side of these inequalities are coordinates of the vector

$M\begin{pmatrix}\delta_{1}\\\ \vdots\\\ \delta_{r}\end{pmatrix},$

where the matrix $M$ is defined by

$M:=(\varepsilon_{i}(\xi_{j}))_{1\leq i,j\leq r}.$

By definition, the absolute value of determinant of $M$ is equal to the
regulator of the number field $L$, and is thus non-zero. Hence $M$ is non-
degenerate. Importantly, the matrix $M$ depends only on the number field $L$.
Further, as the value of the regulator is independent of the choice of the $r$
inequalities we picked, it shows that our choice of inequalities is irrelevant
and we obtain the same result given a different choice of $r$ inequalities
from the $r+1$ in (45).

It follows that the solutions to the system of inequalities (45) for
$i=1,\dots,r$ are given by $M^{-1}\mathcal{B}$, where $\mathcal{B}$ is an
$r$-dimensional cube $[-\mathcal{C}_{44}\left(K\right)\log
H_{L}\left(a,\,b,\,c\right),\mathcal{C}_{44}\left(K\right)\log
H_{L}\left(a,\,b,\,c\right)]^{r}$. Thus these solutions form a parallepiped,
the form of which depends on $M$ (hence eventually on $L$ only, which is
uniquely determined by $K$) and the linear size is given by
$\mathcal{C}_{45}\left(K\right)\log H_{L}(a,\,b,\,c)$. This means that the
solutions $\delta_{i}$ have an upper bound of the form
$\mathcal{C}_{46}\mathcal{C}_{45}\log H(a,\,b,\,c)$, where the constant
$\mathcal{C}_{46}$ depends on $M$ (hence actually depends on $K$) only. We
thus conclude that

$\max_{i}\left|\delta_{i}\right|\leq\mathcal{C}_{47}\left(K\right)\log
H_{L}(a,\,b,\,c),$

as claimed. ∎

Importantly, as commented during the proof, the method is not changed if we
choose a different unit such as $\frac{u_{a}}{u_{b}}$ and so on. Thus this
lemma holds for all relevant units in this paper.

We return to considering (3). We can now use the above after writing the unit
in terms of fundamental units as follows:

$\displaystyle\text{ord}_{\mathfrak{p}}\left(c\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{u_{a}}{u_{b}}a_{1}^{e_{1}}\cdots
a_{t}^{e_{t}}b_{1}^{f_{1}}\dots b_{u}^{f_{u}}-1\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\mu\xi_{1}^{\delta_{1}}\cdots\xi_{r}^{\delta_{r}}a_{1}^{e_{1}}\cdots
a_{t}^{e_{t}}b_{1}^{f_{1}}\cdots b_{u}^{f_{u}}-1\right)$ (46)

where $\mu$ is a root of unity. We are in a position to apply Lemma 3. Using
the notation of Lemma 3, from (41) and Lemma 7 we obtain that $\log
B\leq\mathcal{C}_{42}\left(K\right)\log\log H_{L}(a,\,b,\,c)$.

Applying Lemma 3 on (3), we obtain that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(c\right)<$
$\displaystyle\mathcal{C}_{48}\left(K\right)^{r+t+u+2}\left(r+t+u+1\right)^{5/2}\log\left(2d\left(r+t+u+1\right)\right)\frac{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)}{\left(\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right)^{2}}$
$\displaystyle h^{\prime}(\mu)h^{\prime}(\xi_{1})\cdots
h^{\prime}(\xi_{r})h^{\prime}(a_{1})\cdots
h^{\prime}(a_{t})h^{\prime}(b_{1})\cdots h^{\prime}(b_{u})\log\log
H_{L}(a,\,b,\,c).$ (47)

Note that $h\left(\mu\right)=0$ as $\mu$ is a root of unity so
$h^{\prime}(\mu)=\frac{1}{16e^{2}d^{2}}$, which we take into the constant.
Further, we recall our system of fundamental units satisfies (9) so we take
$\prod_{i=1}^{r}h^{\prime}(\xi_{i})$ into the constant.

We further note that
$\left|{\text{N}_{L/\mathbb{Q}}}\left(a_{i}\right)\right|=\text{Nm}_{\mathbb{Q}}^{L}\left(\left<a_{i}\right>\right)=\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)^{f_{K}}$,
and similarly for $b_{j}$ and $c_{k}$ [18]. We recall that by definition,
$f_{K}\leq d$ [25]. Further, the norms of all these prime ideals are greater
than $1$, so for all
$x\in\left\\{a_{1},\dots,\,a_{s},\,b_{1},\dots,\,b_{t}\right\\},$ if
$\mathfrak{a}_{x}$ is the prime ideal associated with $x$,
$h^{\prime}(x)\leq\mathcal{C}_{49}\left(K\right)\log\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{a}_{x}\right)$.
Putting this together with the inequality above, with other bounds used as
necessary, we obtain that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(c\right)\leq$
$\displaystyle\mathcal{C}_{50}\left(K\right)^{t+u}\left(r+t+u+1\right)^{7/2}\log\log
H_{L}(a,\,b,\,c)$
$\displaystyle\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right).$
(48)

Similarly, we see that by considering (3) in the same way as above, we obtain
that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(b\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{u_{c}}{u_{a}}c_{1}^{g_{1}}\cdots
c_{v}^{g_{v}}a_{1}^{-e_{1}}\dots a_{t}^{-e_{t}}-1\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\mu^{\prime}\xi_{1}^{\delta^{\prime}_{1}}\cdots\xi_{r}^{\delta^{\prime}_{r}}c_{1}^{g_{1}}\cdots
c_{v}^{g_{v}}a_{1}^{-e_{1}}\dots a_{t}^{-e_{t}}-1\right).$ (49)

Following the same line of reasoning as above we obtain that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(b\right)\leq$
$\displaystyle\mathcal{C}_{51}\left(K\right)^{t+v}\left(r+t+v+1\right)^{7/2}\log\log
H_{L}(a,\,b,\,c)$
$\displaystyle\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{j}\right)\right).$
(50)

In the same way, by considering (3) we further obtain that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(a\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(-\frac{u_{b}}{u_{c}}b_{1}^{f_{1}}\cdots
b_{u}^{f_{u}}c_{1}^{-g_{1}}\dots c_{v}^{-g_{v}}-1\right)$
$\displaystyle=\text{ord}_{\mathfrak{p}}\left(\mu^{\prime\prime}\xi_{1}^{\delta^{\prime}_{1}}\cdots\xi_{r}^{\delta^{\prime}_{r}}b_{1}^{f_{1}}\cdots
b_{u}^{f_{u}}c_{1}^{-g_{1}}\dots c_{v}^{-g_{v}}-1\right),$ (51)

and as before, applying Lemma 3 gives us that

$\displaystyle\text{ord}_{\mathfrak{p}}\left(a\right)\leq$
$\displaystyle\mathcal{C}_{52}\left(K\right)^{u+v}\left(r+u+v+1\right)^{7/2}\log\log
H_{L}(a,\,b,\,c)$
$\displaystyle\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\prod_{i=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{i}\right)\right)\cdot\prod_{j=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{j}\right)\right).$
(52)

From this point, all constants depend on the field $K$, in particular on the
degree of the field $d$, so we omit these dependencies.

By combining (35) and (3) we obtain that

$\displaystyle\frac{\log H_{L}(a,\,b,\,c)}{\log\log H_{L}(a,\,b,\,c)}<$
$\displaystyle\mathcal{C}_{53}^{t+u}\left(r+t+u+1\right)^{7/2}\log
G\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
$\displaystyle\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right).$
(53)

Similarly, combining (36) and (3) gives us that

$\displaystyle\frac{\log H_{L}(a,\,b,\,c)}{\log\log H_{L}(a,\,b,\,c)}<$
$\displaystyle\mathcal{C}_{54}^{t+v}\left(r+t+v+1\right)^{7/2}\log
G\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
$\displaystyle\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{j}\right)\right)\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(54)

Applying the same idea, combining (37) and (3) gives us that

$\displaystyle\frac{\log H_{L}(a,\,b,\,c)}{\log\log H_{L}(a,\,b,\,c)}<$
$\displaystyle\mathcal{C}_{55}^{u+v}\left(r+u+v+1\right)^{7/2}\log
G\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\cdot\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}$
$\displaystyle\prod_{i=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{j}\right)\right)\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(55)

Multiplying together (3), (3) and (3) and bounding some terms for ease, we
obtain that

$\displaystyle\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{3}<$
$\displaystyle\mathcal{C}_{56}^{t+u+v}\left(r+t+u+v\right)^{21/2}\left(\log
G\right)^{3}$
$\displaystyle\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)$
$\displaystyle\left(\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)\cdot\prod_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)\right)^{3}$
$\displaystyle\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(56)

We note that $r$ depends on the field, so we can write
$\left(r+\left(t+u+v\right)\right)^{21/2}\leq\mathcal{C}_{57}\left(t+u+v\right)^{21/2}$.
Further, for sufficiently large $\mathcal{C}_{57}$ this will absorb
$\left(t+u+v\right)^{21/2}$, so we can move this into the constant. We thus
obtain that

$\displaystyle\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{3}<$
$\displaystyle\mathcal{C}_{58}^{t+u+v}\left(\log
G\right)^{3}\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)$
$\displaystyle\left(\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)\cdot\prod_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)\right)^{3}$
$\displaystyle\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(57)

Next we aim to deal with
$\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)\cdot\prod_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)$.
First note that
$\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{P}\right)^{h_{K}}=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{P}\mathcal{O}_{L}\right)$,
where $\mathfrak{P}$ is a prime ideal of $\mathcal{O}_{K}$. We follow an idea
from the first part of Section 3 of [34]. Let $N$ be the number of prime
ideals of $\mathcal{O}_{L}$ such that the prime ideal
$\mathfrak{P}\mid\left(abc\right)\mathcal{O}_{L}$. By definition, these all
lie above primes $\mathfrak{p}$ of $\mathcal{O}_{K}$, so $N\geq t+u+v$. Thus
from these comments and Lemma 4 we obtain that

$\displaystyle\left(\frac{t+u+v}{\mathcal{C}_{59}}\right)^{t+u+v}\leq\left(\frac{N}{\mathcal{C}_{60}}\right)^{N}<G,$
(58)

where $\mathcal{C}_{60}$ is the constant given by Lemma 4. It follows that

$t+u+v<\mathcal{C}_{61}\frac{\log G}{\log\log G}.$ (59)

By the arithmetic-geometric mean inequality we obtain that

$\displaystyle\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)\cdot\prod_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{t+u+v}\left(\sum_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)+\sum_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)+\sum_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)\right)\right)^{t+u+v}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{t+u+v}\sum_{\begin{subarray}{c}\mathfrak{P}\subset\mathcal{O}_{L}\\\
\mathfrak{P}\text{prime}\\\
\mathfrak{P}\mid\left(abc\right)\mathcal{O}_{L}\end{subarray}}\log\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{P}\right)\right)\right)^{t+u+v}$
$\displaystyle\leq$ $\displaystyle\left(\frac{\log G}{t+u+v}\right)^{t+u+v}.$
(60)

It follows from (59) and (3) that

$\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)\cdot\prod_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)<G^{\mathcal{C}_{62}\frac{\log\log\log
G}{\log\log G}}.$ (61)

The same logic can be used to show that

$\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\ \mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)<G^{\mathcal{C}_{63}\frac{\log\log\log
G}{\log\log G}},$ (62)

and that

$\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\ \mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)<G^{\mathcal{C}_{64}\frac{\log\log\log
G}{\log\log G}}.$ (63)

Applying these to (3), we obtain that

$\displaystyle\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{3}<$
$\displaystyle\mathcal{C}_{58}^{t+u+v}\left(\log G\right)^{3}$
$\displaystyle\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)G^{\mathcal{C}_{65}\frac{\log\log\log
G}{\log\log G}}.$ (64)

Further, from Lemma 4 we obtain that

$\mathcal{C}_{58}^{t+u+v}<G^{\frac{\mathcal{C}_{66}}{\log\log G}}.$ (65)

Further, note that $\log G=G^{\frac{\log\log G}{\log G}}$. Thus we obtain that

$\displaystyle\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{3}<$
$\displaystyle\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)$
$\displaystyle G^{\mathcal{C}_{67}\left(\frac{\log\log\log G}{\log\log
G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (66)

We take the cube root of both sides, before applying Lemma 5, obtaining

$\displaystyle\log H_{L}(a,\,b,\,c)<$
$\displaystyle\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)^{\frac{1}{3}}$
$\displaystyle G^{\mathcal{C}_{68}\left(\frac{\log\log\log G}{\log\log
G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (67)

Note, the dominant term in the power of $G$ is $\frac{\log\log\log G}{\log\log
G}$. Combining this with the above proves Theorem 1.

###  3.1 Corollaries of Theorem 1

In this section we show various corollaries of Theorem 1. The first two
corollaries depend on the Class Group of $K$ and the ideals that
$\mathfrak{p}_{b}$ and $\mathfrak{p}_{c}$ lie above.

######  Corollary 1.

Assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)>\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
and that $\mathfrak{p}_{b}$ and $\mathfrak{p_{c}}$ both lie over prime ideals
of $\mathcal{O}_{K}$ that do not generate the class group of $K$. Then

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<G^{\frac{1}{3}+\mathcal{C}_{69}\frac{\log\log\log G}{\log\log
G}}.$ (68)

###### Proof.

By assumption,

$\displaystyle\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)$
$\displaystyle=\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)^{2}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right).$

Recall that in the Hilbert Class Field $L$, a prime ideal $\mathfrak{p}$ of
$\mathcal{O}_{K}$ splits into $\frac{h_{K}}{P}$ prime ideals of
$\mathcal{O}_{L}$, where $P$ is the order of $\left[\mathfrak{p}\right]$ in
the Class Group of $K$. By assumption, there must be at least two prime ideals
dividing $b\mathcal{O}_{L}$ with the same norm
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)$, and similarly for
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$.

It follows that

$\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)^{2}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right)\leq
G,$

and the claim follows. ∎

######  Corollary 2.

Assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)<\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
and that $\mathfrak{p}_{c}$ lies above a prime ideal of $\mathcal{O}_{K}$ that
has order greater than 2 in the class group of $K$. Then

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<G^{\frac{1}{3}+\mathcal{C}_{70}\frac{\log\log\log G}{\log\log
G}}.$ (69)

###### Proof.

By assumption,

$\displaystyle\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}\right)$
$\displaystyle=\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{3}\right).$

By the comments in the proof of the previous corollary, our assumption here
gives us that there are at least 3 prime ideals of $\mathcal{O}_{L}$ dividing
$c\mathcal{O}_{L}$ with the same norm,
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$. It follows that

$\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{3}\right)\leq
G,$

and the claim follows. ∎

The following corollary holds regardless of the class field of $K$.

######  Corollary 3.

Assume that
$\textrm{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)>\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$.
Then

$\log H_{L}(a,\,b,\,c)<G^{\frac{2}{3}+\mathcal{C}_{71}\frac{\log\log\log
G}{\log\log G}}.$

###### Proof.

Note that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G$. Further, by assumption

$\displaystyle\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}$
$\displaystyle=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
$\displaystyle\leq G.$

Thus the corollary follows. ∎

######  Corollary 4.

Assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)>\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)>\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right).$
Then

$\log H_{L}(a,\,b,\,c)<G^{\frac{5}{9}+\mathcal{C}_{72}\frac{\log\log\log
G}{\log\log G}}.$

If
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
then we obtain that

$\log H_{L}(a,\,b,\,c)<G^{\frac{2}{3}+\mathcal{C}_{72}\frac{\log\log\log
G}{\log\log G}}.$

###### Proof.

By assumption,
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G^{\frac{2}{3}}$ and
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\leq
G^{\frac{1}{2}},\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G^{\frac{1}{3}}$. Applying this to Theorem 1 gives both parts of the
corollary. ∎

######  Remarks.

If we assume that none of $a,\,b,\,c$ are units of $\mathcal{O}_{K}$ then the
only assumption we need to obtain the first inequality in the corollary above
is that
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)$.
This follows as then by assumption,
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\leq
G^{\frac{1}{3}},\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G^{\frac{1}{3}}$. The argument follows.

We now present some corollaries that depend on the value of
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}$.

######  Corollary 5.

Assume that
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}<G^{\alpha}$
with $0<\alpha\leq 1$. Then

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1+2\alpha}{3}+\mathcal{C}_{73}\frac{\log\log\log
G}{\log\log G}}.$

###### Proof.

Again,

$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G.$

By assumption,

$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}<G^{2\alpha}.$

Applying this to Theorem 1 gives the result. ∎

######  Corollary 6.

Assume that
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}<\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}$ for $0<\alpha<\frac{2}{3}$. Then

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1}{3-2\alpha}+\mathcal{C}_{74}\frac{\log\log\log
G}{\log\log G}}.$

###### Proof.

Consider (3). Applying the assumption, we can rewrite this as

$\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{3}<G\left(\log
H_{L}(a,\,b,\,c)\right)^{2\alpha}G^{\mathcal{C}_{75}\frac{\log\log\log
G}{\log\log G}}.$

Dividing through by $\left(\log H_{L}(a,\,b,\,c)\right)^{2\alpha}$ we obtain
that

$\frac{\left(\log H_{L}(a,\,b,\,c)\right)^{3-2\alpha}}{\left(\log\log
H_{L}(a,\,b,\,c)\right)^{3}}<G^{1+\mathcal{C}_{75}\frac{\log\log\log
G}{\log\log G}}.$

Taking the $3-2\alpha$’th root and applying a variant of Lemma 5 gives the
result. ∎

######  Corollary 7.

Assume that
$\textrm{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{max}\right)<\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}$ for $0<\alpha<\frac{3}{5}$. Then

$\displaystyle\log H_{L}\left(a,\,b,\,c\right)$
$\displaystyle<G^{\frac{\mathcal{C}_{76}}{3-5\alpha}\frac{\log\log\log
G}{\log\log G}}$ $\displaystyle=G^{\mathcal{C}_{77}\frac{\log\log\log
G}{\log\log G}}.$ (70)

######  Remarks.

We note that we can write this in the following terms. For any given
$\varepsilon>0$, given the assumptions of the theorem and corollary there is a
computable number $\mathcal{C}_{78}$ such that

$\log H_{L}\left(a,\,b,\,c\right)<G^{\mathcal{C}_{78}\cdot\varepsilon}.$

###### Proof.

Consider Theorem 1. By assumption,

$\log H_{L}\left(a,\,b,\,c\right)<\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\frac{5\alpha}{3}}G^{\frac{\log\log\log
G}{\log\log G}}.$

∎

Dividing through, we obtain that

$\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{1-\frac{5\alpha}{3}}<G^{\frac{\log\log\log
G}{\log\log G}}.$

Take the $\frac{1}{1-\frac{5\alpha}{3}}$’th root and the result follows.

##  4 Method Only Using two $S$-unit bounds

Part of the difficulty in analysing cases in the previous section comes from
the number of prime ideals on the right hand side of (3). If we only use two
$S$-unit bounds then, while in general the bound is worse, it is easier to
analyse for corollaries. We now prove Theorem 2, as stated in the
introduction.

We follow the main text until (3). We then do not use the $S$-unit bound
obtained by letting $S$ be equal to the infinite places and finite places
corresponding to the prime ideals of $\mathcal{O}_{L}$ dividing
$bc\mathcal{O}_{L}$. Following the argument of the main text, we obtain (3)
and (3). Multiplying these together we obtain that

$\displaystyle\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{2}<\mathcal{C}_{79}^{t+u+v}\left(r+t+u+v\right)^{7}\left(\log
G\right)^{2}\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right)$
$\displaystyle\left(\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right)\cdot\prod_{k=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{k}\right)\right)\right)^{2}\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(71)

From here we follow the arguments of the proof of the main theorem, obtaining

$\left(\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}\right)^{2}<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right)G^{\mathcal{C}_{80}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (72)

We take the square root of both sides, before applying Lemma 5, obtaining

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right)^{\frac{1}{2}}G^{\mathcal{C}_{81}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}$
$\displaystyle=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)^{\frac{1}{2}}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)G^{\mathcal{C}_{81}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (73)

Again, $\frac{\log\log\log G}{\log\log G}$ is the dominant term in the
exponent of $G$.

From this point, there are many corollaries we can find, similarly to in the
previous section. However, given that there are fewer prime ideal on the right
hand side of (4), they are generally easier to prove. Further, Theorem 1 and
Theorem 2 are independent, so if
$\textrm{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)$ is sufficiently
large in comparison to
$\textrm{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)$ and
$\textrm{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$, Theorem 2 could
give a better bound.

###  4.1 Corollaries Of Theorem 2

This corollary relies on the class group of $\mathcal{O}_{K}$.

######  Corollary 8.

Assume that the prime ideal $\mathfrak{r}\subset\mathcal{O}_{K}$ that
$\mathfrak{p}_{c}\subset\mathcal{O}_{L}$ lies above does not generate the
class group of $K$. Then there exits an effectively computable constant
$\mathcal{C}_{82}$ such that

$\log H_{L}(a,\,b,\,c)<G^{\frac{1}{2}+\mathcal{C}_{82}\frac{\log\log\log
G}{\log\log G}}.$ (74)

###### Proof.

Let $\mathfrak{r}$ be a prime ideal of $\mathcal{O}_{K}$ dividing
$c\mathcal{O}_{K}$ such that $\mathfrak{p_{c}}\subset\mathcal{O}_{L}$ lies
above $\mathfrak{r}$. Assume that $\mathfrak{r}$ does not generate the class
group of $K$. Then in $L=HCF(K)$, $\mathfrak{r}$ splits into $h_{K}/P$ prime
ideals, where $h_{K}$ is the class number of $K$ and $P$ is the order of
$\left[\mathfrak{r}\right]$ in $C_{K}$ [7] [25]. As $\mathfrak{r}$ does not
generate the class group of $K$, then the order of $\left[\mathfrak{r}\right]$
is at least 2. As $\mathfrak{r}$ splits in $\mathcal{O}_{L}$, we know that all
prime ideals lying above $\mathfrak{r}$ in $\mathcal{O}_{L}$ have the same
norm. By assumption, we have at least two such ideals in $\mathcal{O}_{L}$, so
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right)^{2}<G$.
More explicitly, there is another prime ideal $\mathfrak{p}_{c}^{\prime}$ of
$\mathcal{O}_{L}$ lying above $\mathfrak{r}$ such that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)=\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)$.
It then follows from Theorem 2 that

$\log H_{L}(a,\,b,\,c)<G^{\frac{1}{2}+\mathcal{C}_{82}\frac{\log\log\log
G}{\log\log G}}.$ (75)

∎

The following corollaries give different bounds depending on
$\mathfrak{p}_{max}$ or $\max\text{ord}_{\mathfrak{p}}\left(c\right)$.

######  Corollary 9.

Assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)<G^{\alpha}$ with
$0<\alpha<1$, or that $\max\text{ord}_{\mathfrak{p}}\left(c\right)<G^{\alpha}$
with $0<\alpha<1$. Then there exits an effectively computable constant
$\mathcal{C}_{83}$ such that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1+\alpha}{2}+\mathcal{C}_{83}\frac{\log\log\log
G}{\log\log G}}.$ (76)

Further, if
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}<G^{\alpha}$
then

$\log H_{L}(a,\,b,\,c)<G^{\frac{3\alpha}{2}+\mathcal{C}_{84}\frac{\log\log\log
G}{\log\log G}}.$ (77)

######  Remarks.

We note that $\frac{3\alpha}{2}<1$ for $\alpha<\frac{2}{3}$, and further that
$\frac{3\alpha}{2}<\frac{1+\alpha}{2}$ for $\alpha<\frac{1}{2}$. Thus, our
second bound is better than our first given in this corollary for
$\alpha<\frac{1}{2}$.

###### Proof.

We first assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)<G^{\alpha}$ where
$\alpha\in(0,\,1)$. Thus

$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\text{b}}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\text{c}}\right)^{2}<G^{1+\alpha}<G^{2}.$

Thus from Theorem 2, we obtain that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1+\alpha}{2}+\mathcal{C}_{83}\frac{\log\log\log
G}{\log\log G}}.$ (78)

We now assume that $\max\text{ord}_{\mathfrak{p}}\left(c\right)<G^{\alpha}$
for some $\alpha\in(0,\,1)$. Then, in place of (3), we have that for all
$\mathfrak{p}\mid c\mathcal{O}_{L}$,
$\text{ord}_{\mathfrak{p}}\left(c\right)<G^{\alpha}$. It follows from (35)
that

$\log H_{L}(a,\,b,\,c)<\mathcal{C}_{84}\left(\log G\right)G^{\alpha}.$ (79)

We note that for $\alpha<\frac{1}{2}$, this bound is actually better than the
bound that follows.

As in the proof for the main theorem, (3) still holds. Multiplying the above
and (3) we obtain that

$\displaystyle\frac{\left(\log H_{L}(a,\,b,\,c)\right)^{2}}{\log\log
H_{L}(a,\,b,\,c)}<$
$\displaystyle\mathcal{C}_{85}^{t+v}\left(r+t+v+1\right)^{\frac{7}{2}}\left(\log
G\right)^{2}G^{\alpha}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
$\displaystyle\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{j}\right)\right)\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(80)

We note that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G$. Further, we can use the techniques from above to tidy terms in the same
way as we did for the main theorem to obtain

$\frac{\left(\log H_{L}(a,\,b,\,c)\right)^{2}}{\log\log
H_{L}(a,\,b,\,c)}<G^{1+\alpha}G^{\mathcal{C}_{86}\frac{\log\log\log
G}{\log\log G}}.$ (81)

Taking the square root and applying a variant of Lemma 5, we obtain that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1+\alpha}{2}+\mathcal{C}_{83}\frac{\log\log\log
G}{\log\log G}}.$ (82)

This proves the first part of Corollary 9. The further comments follow
directly from Theorem 2 when we bound
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)$ and
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$ above by
$G^{\alpha}$. This gives us that

$\log H_{L}(a,\,b,\,c)<G^{\frac{3\alpha}{2}+\mathcal{C}_{8}4\frac{\log\log\log
G}{\log\log G}},$ (83)

as claimed, where $\frac{3\alpha}{2}<1$ for $\alpha<\frac{2}{3}$, and is a
better bound than given above for $\alpha<\frac{1}{2}$.

It also follows directly from Theorem 2 that if
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\text{b}}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\text{c}}\right)\right\\}<G^{\alpha}$,
then

$\log H_{L}(a,\,b,\,c)<G^{\frac{3\alpha}{2}+\mathcal{C}_{84}\frac{\log\log\log
G}{\log\log G}}.$

∎

######  Corollary 10.

Assume now that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)<\left(\log
H_{L}(a,\,b,\,c)\right)^{\alpha}$ with $0<\alpha<1$, or that
$\max\text{ord}_{\mathfrak{p}}\left(c\right)<\left(\log
H_{L}(a,\,b,\,c)\right)^{\alpha}$ with $0<\alpha<1$. Then there exits an
effectively computable constant $\mathcal{C}_{87}$ such that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1}{2-\alpha}+\mathcal{C}_{87}\frac{\log\log\log
G}{\log\log G}}.$ (84)

Furthermore, if
$\max\left\\{\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right),\,\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\right\\}<\left(\log
H_{L}(a,\,b,\,c)\right)^{\alpha}$ for $\alpha<\frac{2}{3}$, then directly from
Theorem 2 we obtain that

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<G^{\frac{\mathcal{C}_{88}}{2-3\alpha}\frac{\log\log\log
G}{\log\log G}}$ $\displaystyle=G^{\mathcal{C}_{89}\frac{\log\log\log
G}{\log\log G}}.$ (85)

This is the best bound we achieve in this text.

######  Remarks.

We note that the second inequality in this corollary gives a sub-exponential
bound, an improvement on the bounds given in [33][34].

To more easily compare with existing results, we note we can slightly weaken
this upper bound. Inequality (10) implies that given any $\varepsilon>0$ there
exists some computable $\mathcal{C}_{90}$ such that

$\log H_{L}\left(a,\,b,\,c\right)<G^{\mathcal{C}_{90}\cdot\epsilon},$

where importantly $\mathcal{C}_{90}$ does not depend on $\epsilon$.

###### Proof.

We first assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)<\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}$ with $\alpha\in(0,1)$. Then from
this assumption and (72), we obtain that

$\frac{\left(\log H_{L}(a,\,b,\,c)\right)^{2-\alpha}}{\left(\log\log
H_{L}(a,\,b,\,c)\right)^{2}}<G\cdot G^{\mathcal{C}_{91}\frac{\log\log\log
G}{\log\log G}}.$ (86)

By assumption, $2-\alpha>1$, and we take this root to obtain that

$\frac{\log H_{L}(a,\,b,\,c)}{\left(\log\log
H_{L}(a,\,b,\,c)\right)^{\frac{2}{2-\alpha}}}<G^{\frac{1}{2-\alpha}+\mathcal{C}_{92}\frac{\log\log\log
G}{\log\log G}}.$ (87)

Note that $1>\frac{1}{2-\alpha}>\frac{1}{2}$. Applying a variant of Lemma 5,
we obtain that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1}{2-\alpha}+\mathcal{C}_{93}\frac{\log\log\log
G}{\log\log G}}.$ (88)

Now instead assume that
$\max\text{ord}_{\mathfrak{p}}\left(c\right)<\left(\log
H_{L}(a,\,b,\,c\right)^{\alpha}$ for some $\alpha\in(0,\,1)$. Then as above,
from (35) we obtain that

$\log H_{L}(a,\,b,\,c)<\mathcal{C}_{94}\log G\left(\log
H_{L}(a,\,b,\,c)\right)^{\alpha}.$ (89)

It immediately follows that

$\left(\log H_{L}(a,\,b,\,c)\right)^{1-\alpha}<\mathcal{C}_{94}\log G.$ (90)

Again, we still have (3), as obtained by following the main argument. We
multiply (3) by the above to obtain

$\displaystyle\frac{\left(\log H_{L}(a,\,b,\,c)\right)^{2-\alpha}}{\log\log
H_{L}(a,\,b,\,c)}<$
$\displaystyle\mathcal{C}_{95}^{t+v}\left(r+t+v+1\right)^{\frac{7}{2}}\left(\log
G\right)^{2}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$
$\displaystyle\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{v}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{r}_{j}\right)\right)\cdot\prod_{\begin{subarray}{c}\mathfrak{p}\in\mathcal{O}_{L}\\\
\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right).$
(91)

As before, we can use the same method of tidying as in the proof of the main
theorem to show that

$\frac{\left(\log H_{L}(a,\,b,\,c)\right)^{2-\alpha}}{\log\log
H_{L}(a,\,b,\,c)}<G^{1+\mathcal{C}_{96}\frac{\log\log\log G}{\log\log G}}.$
(92)

Taking the $2-\alpha$’th root and applying a variant of Lemma 5, we obtain
that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{1}{2-\alpha}+\mathcal{C}_{97}\frac{\log\log\log
G}{\log\log G}}.$ (93)

This concludes the proof of the first part of Corollary 10.

To see the strongest case, we appeal directly to Theorem 2. Assume that

$\text{Nm}\left(\mathfrak{p}_{\max}\right)<\log\left(H_{L}(a,\,b,\,c)\right)^{\alpha}$

with $\alpha<\frac{2}{3}$. Then we can bound
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)$ and
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$ above by
$\log\left(H_{L}(a,\,b,\,c)\right)^{\alpha}$, obtaining

$\log H_{L}(a,\,b,\,c)<\left(\log
H_{L}(a,\,b,\,c)\right)^{\frac{3\alpha}{2}}G^{\mathcal{C}_{98}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (94)

It then follows that

$\left(\log
H_{L}(a,\,b,\,c)\right)^{1-\frac{3\alpha}{2}}<G^{\mathcal{C}_{98}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (95)

Taking the $\frac{1}{1-\frac{3\alpha}{2}}$th root gives us the result, namely
that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{\mathcal{C}_{98}}{2-3\alpha}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}.$ (96)

Finally, we recall that the dominant term in the exponent is
$\frac{\log\log\log G}{\log\log G}$, so we obtain that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{\mathcal{C}_{88}}{2-3\alpha}\left(\frac{\log\log\log
G}{\log\log G}\right)}.$ (97)

As commented in the statement of the theorem, this is of the form

$\log H_{L}(a,\,b,\,c)<G^{\mathcal{C}_{90}\cdot\epsilon}.$

∎

######  Remarks.

While the assumptions are hard to compare due to their different natures, we
can see that for all $\alpha\in(0,\,1)$,
$\frac{1+\alpha}{2}\geq\frac{1}{2-\alpha}$. Thus, generally speaking, the
bound of Corollary 10 is better than that of Corollary 9. More concretely,
given $(a,\,b,\,c)$ that satisfy the assumptions of both Corollary 9 and 10,
Corollary 10 gives a better bound in terms of the radical $G$ than that of
Corollary 9.

######  Corollary 11.

Assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)>G^{\alpha}$ for
$\alpha>\frac{1}{3}$, and that $\mathfrak{p}_{a}=\mathfrak{p}_{\max}$. Then

$\log
H_{L}(a,\,b,\,c)<G^{\frac{3-3\alpha}{2}+\mathcal{C}_{99}\frac{\log\log\log
G}{\log\log G}}.$ (98)

If $\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)\leq
G^{\frac{1}{3}}$ it follows directly from Theorem 2 that

$\log H_{L}(a,\,b,\,c)<G^{\frac{1}{2}+\mathcal{C}_{100}\frac{\log\log\log
G}{\log\log G}}.$ (99)

######  Remarks.

Note we have the assumption that $\alpha>\frac{1}{3}$ in order to make sure
that $\frac{3-3\alpha}{2}<1$.

Further, the second inequality given is the same case as $\alpha=\frac{1}{3}$
in the last part of Corollary 9.

###### Proof.

Assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)>G^{\alpha}$, and
assume that $\mathfrak{p}_{a}=\mathfrak{p}_{\max}$. Then considering (4), we
note that

$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}<\left(G^{1-\alpha}\right)^{3},$

so

$\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{2}\right)^{\frac{1}{2}}<G^{\frac{3-3\alpha}{2}}.$

It follows that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{3-3\alpha}{2}+\mathcal{C}_{99}\frac{\log\log\log
G}{\log\log G}}.$ (100)

We note that $\frac{3-3\alpha}{2}<1$ for $\alpha>\frac{1}{3}$. ∎

######  Corollary 12.

Assume $\text{ord}_{\mathfrak{p}_{c}}c<G^{\alpha}$ for $0<\alpha\leq 1$. Then

$\log
H_{L}\left(a,\,b,\,c\right)<G^{\max\left\\{\alpha,\,\frac{3}{4}\right\\}+\mathcal{C}_{101}\frac{\log\log\log
G}{\log\log G}}.$ (101)

###### Proof.

Assume that $\text{ord}_{\mathfrak{p}_{c}}c<G^{\alpha}$.

Note that we can write

$\max_{\mathfrak{p}\mid\left<c\right>_{L}}\text{ord}_{\mathfrak{p}}\left(c\right)=\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right),\,\text{ord}_{\mathfrak{p}_{c}}\left(c\right)\right\\}.$

By assumption, we attain the bound

$\max_{\mathfrak{p}\mid\left<c\right>_{L}}\text{ord}_{\mathfrak{p}}\left(c\right)=\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right),\,G^{\alpha}\right\\}.$
(102)

From the above and (35), it follows directly that

$\log H_{L}\left(a,\,b,\,c\right)<\mathcal{C}_{102}\log
G\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right),\,G^{\alpha}\right\\}.$
(103)

We now consider cases depending on
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$.

First assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)<G^{\frac{1}{2}}$.
Then we can directly use Corollary 9 to obtain the bound given there, namely

$\log H_{L}(a,\,b,\,c)<G^{\frac{3}{4}+\mathcal{C}_{103}\frac{\log\log\log
G}{\log\log G}}.$ (104)

Thus from the above and (103), we obtain that

$\log
H_{L}\left(a,\,b,\,c\right)<G^{\max\left\\{\frac{3}{4},\,\alpha\right\\}+\mathcal{C}_{104}\frac{\log\log\log
G}{\log\log G}}$ (105)

Assume now instead that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)\geq
G^{\frac{1}{2}}$. It immediately follows that for all other prime ideals
$\mathfrak{p}$ contributing to $G$, we have that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)<G^{\frac{1}{2}}$.

Consider now $\max_{\begin{subarray}{c}\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right)$.
We apply Yu’s bound as before on this, taking the above comments into
consideration. It follows that

$\displaystyle\max_{\begin{subarray}{c}\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right)<$
$\displaystyle\mathcal{C}_{105}^{t+u}\left(r+t+u+1\right)^{7/2}\log\log
H_{L}(a,\,b,\,c)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)$
$\displaystyle\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right),$
(106)

where $\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)<G^{\frac{1}{2}}$.
Following the same logic as the main text, we obtain that

$\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}<\mathcal{C}_{106}^{t+u}\left(r+t+u+1\right)^{7/2}\log G\cdot
G^{\frac{1}{2}}\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right).$
(107)

Note that (3) still holds, and
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\leq
G$. Multiplying (3) and (107), tidying terms as we do in the text, and
considering (103), we obtain that

$\displaystyle\frac{\log H_{L}\left(a,\,b,\,c\right)}{\log\log
H_{L}\left(a,\,b,\,c\right)}<$
$\displaystyle\max\left\\{G^{\frac{3}{4}+\mathcal{C}_{107}\frac{\log\log\log
G}{\log\log G}},\,G^{\alpha+\mathcal{C}_{108}\frac{\log\log\log G}{\log\log
G}}\right\\}.$ (108)

More concisely, after applying Lemma 5, we obtain that

$\log
H_{L}\left(a,\,b,\,c\right)<G^{\max\left\\{\alpha,\,\frac{3}{4}\right\\}+\mathcal{C}_{101}\frac{\log\log\log
G}{\log\log G}}.$ (109)

Thus, in either case depending on
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)$, we obtain that

$\log
H_{L}\left(a,\,b,\,c\right)<G^{\max\left\\{\alpha,\,\frac{3}{4}\right\\}+\mathcal{C}_{101}\frac{\log\log\log
G}{\log\log G}}.$ (110)

∎

######  Corollary 13.

Assume that $\text{ord}_{\mathfrak{p}_{c}}c<\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}$ for $0<\alpha<1$. Then

$\log
H_{L}\left(a,\,b,\,c\right)<\max\left\\{G^{\frac{3}{4}+\mathcal{C}_{109}\left(\frac{\log\log\log
G}{\log\log G}\right)},\,\mathcal{C}_{110}\left(\log
G\right)^{\frac{1}{1-\alpha}}\right\\}.$ (111)

###### Proof.

Assume that $\text{ord}_{\mathfrak{p}_{c}}(c)<\left(\log
H_{L}(a,\,b,\,c)\right)^{\alpha}$ for some $0<\alpha<1$. As in Corollary 12 it
immediately follows that

$\displaystyle\max_{\mathfrak{p}\mid\left<c\right>_{L}}\text{ord}_{\mathfrak{p}}\left(c\right)$
$\displaystyle=\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right),\,\text{ord}_{\mathfrak{p}_{c}}\left(c\right)\right\\}$
$\displaystyle\leq\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right),\,\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}\right\\}.$ (112)

This along with (35) implies that

$\log H_{L}(a,\,b,\,c)<\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right)\log
G,\,\mathcal{C}_{111}\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}\log G\right\\}.$ (113)

If

$\max\left\\{\max_{\begin{subarray}{c}\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right)\log
G,\,\left(\log H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}\log
G\right\\}=\mathcal{C}_{112}\left(\log
H_{L}\left(a,\,b,\,c\right)\right)^{\alpha}\log G,$

then we can see that

$\log H_{L}(a,\,b,\,c)<\mathcal{C}_{113}\left(\log
G\right)^{\frac{1}{1-\alpha}}.$

We now consider two cases.

In the first case we assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)<G^{\frac{1}{2}}$.
In this case we can appeal directly to Corollary 9, obtaining that

$\log
H_{L}(a,\,b,\,c)<G^{\frac{3}{4}+\mathcal{C}_{114}\left(\frac{\log\log\log
G}{\log\log G}\right)}.$ (114)

For the second case we assume that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{\max}\right)\geq
G^{\frac{1}{2}}$ and follow the same argument as in Case 2 in Corollary 12;
see section 4.5.2.

As before, we see that for all prime ideals
$\mathfrak{p}\neq\mathfrak{p}_{\max}$ contributing to $G$, we have that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)<G^{\frac{1}{2}}$.
Applying Yu’s bound on $\max_{\begin{subarray}{c}\mathfrak{p}\mid
c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right)$
again, we find that

$\max_{\begin{subarray}{c}\mathfrak{p}\mid c\mathcal{O}_{L}\\\
\mathfrak{p}\neq\mathfrak{p}_{c}\end{subarray}}\text{ord}_{\mathfrak{p}}\left(c\right)<\mathcal{C}_{115}^{s+t}\left(r+t+u+1\right)^{7/2}\log\log
H_{L}(a,\,b,\,c)\cdot\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right),$
(115)

Again, we know that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)<G^{\frac{1}{2}}$.
Following the logic of the main text and Section 4.5.2, it again follows that

$\frac{\log H_{L}(a,\,b,\,c)}{\log\log
H_{L}(a,\,b,\,c)}<\mathcal{C}_{116}^{s+t}\left(r+t+u+1\right)^{7/2}\log G\cdot
G^{1-\beta}\prod_{i=1}^{t}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}_{i}\right)\right)\cdot\prod_{j=1}^{u}\log\left(\text{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{q}_{j}\right)\right).$
(116)

After tidying as we have previously and applying Lemma 5, it follows that

$\log H_{L}(a,\,b,\,c)<G^{\frac{3}{4}+\mathcal{C}_{117}\frac{\log\log\log
G}{\log\log G}}.$

Combining these results, in both cases we obtain that

$\log
H_{L}\left(a,\,b,\,c\right)<\max\left\\{G^{\frac{3}{4}+\mathcal{C}_{109}\frac{\log\log\log
G}{\log\log G}},\,\mathcal{C}_{110}\left(\log
G\right)^{\frac{1}{1-\alpha}}\right\\}.$ (117)

∎

##  5 Application of Le Fourn’s Method

A method of Le Fourn [19] allows us to improve the above somewhat. We will
prove Theorem 3 from the introduction. The important point here is that in our
$S$-unit bounds we can use the following lemma to replace the use of the
largest prime with the third largest prime in the set $S$. This reduces our
reliance on $\mathfrak{p}_{a},\,\mathfrak{p}_{b},\,\mathfrak{p}_{c}$.

######  Lemma 8.

Let $K$ be a number field of degree $d$ and let $S\subset M_{K}$ containing
all the infinite places and a finite number of finite places. Let
$s=\left|S\right|$ and let $\alpha,\,\beta\in K^{*}$. Consider the $S$-unit
equation

$\alpha x+\beta y=1$

with $x,\,y\in\mathcal{O}_{S}^{*}$.

If $S$ contains at most two finite places then all solutions of the above
satisfy

$\max\left\\{h\left(x\right),\,h\left(y\right)\right\\}\leq\mathcal{C}_{118}\left(d,\,s\right)R_{S}\log^{+}\left(R_{S}\right)H,$

where
$H=\max\left\\{h\left(\alpha\right),\,h\left(\beta\right),\,1,\,\frac{\pi}{d}\right\\}$,
$R_{S}$ is the $S$-regulator and $\mathcal{C}_{118}(d,\,s)$ is given in [19].

For any general set of places $S$, all solutions of the above equation satisfy

$\max\left\\{h\left(x\right),\,h\left(y\right)\right\\}\leq\mathcal{C}_{119}\left(d,\,s\right)P^{\prime}_{S}R_{S}\left(1+\frac{\log^{+}\left(R_{S}\right)}{\log^{+}\left(P^{\prime}_{S}\right)}\right)H,$

with $P^{\prime}_{S}$ the third largest value of the norms of ideals coming
from the finite places of $S$ and $\mathcal{C}_{119}\left(d,\,s\right)$ given
in [19]. If there are fewer than 3 finite places in $S$ then we take
$P^{\prime}_{S}=1$.

###### Proof.

This is Theorem 1.4 of [19]. We note the constants are taken from [13]. ∎

When we apply Le Fourn’s lemma in the places we previously applied Lemma 2 we
attain (after moving things into the constant) essentially the same bounds
with $\mathfrak{p}_{a},\,\mathfrak{p}_{b},\,\mathfrak{p}_{c}$ replaced by
$\mathfrak{p}_{a}^{\prime},\,\mathfrak{p}_{b}^{\prime}$ and
$\mathfrak{p}_{c}^{\prime}$ where $\mathfrak{p}_{a}^{\prime}$ is the prime
ideal of third largest norm dividing $a\mathcal{O}_{L}$ and similarly for
$\mathfrak{p}_{b}^{\prime}$ and $\mathfrak{p}_{c}^{\prime}$. If fewer than
three prime ideals divide $a,\,b$ or $c$ then we define the corresponding norm
to be 1.

We follow the proof of the main theorem, but we replace any use of Lemma 2
with Lemma 8. For the most part, all that changes is any occurrence of
$\mathfrak{p}_{a},\,\mathfrak{p}_{b}$ and $\mathfrak{p}_{c}$ arising from the
use of Lemma 2 is replaced by
$\mathfrak{p}_{a}^{\prime},\,\mathfrak{p}_{b}^{\prime}$ and
$\mathfrak{p}_{c}^{\prime}$. We follow the line of reasoning from the main
text up until (3). For this first application of $S$-units, where we have no
finite places, we continue to use Lemma (2) as it is simpler in this case than
Lemma 8. As before, we obtain (3).

As before, now let $S=S_{\infty}\cup\left\\{\mathfrak{p}\,:\,\mathfrak{p}\mid
c\mathcal{O}_{L}\right\\}.$ Applying Lemma 8 to

$-\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}-\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}=1,$

we obtain that

$\displaystyle\max\left\\{h\left(-\frac{u_{a}}{u_{c}c^{\prime}}\right),\,h\left(-\frac{u_{b}}{u_{c}}\right)\right\\}$
$\displaystyle<\mathcal{C}_{120}P^{\prime}_{S}R_{S}\left(1+\frac{\log^{+}R_{S}}{\log^{+}P^{\prime}_{S}}\right)\max\left\\{h\left(a^{\prime}\right),\,h\left(b^{\prime}\right),\,1,\,\frac{\pi}{d}\right\\}$
$\displaystyle<\mathcal{C}_{121}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}^{\prime}_{c}\right)R_{S}^{2}\max{\left\\{h\left(a^{\prime}\right),\,h\left(b^{\prime}\right),\,1,\,\frac{\pi}{d}\right\\}}$
$\displaystyle<\mathcal{C}_{122}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}^{\prime}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\subset\mathcal{O}_{L}\\\
\mathfrak{p}\mid
c\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)^{2}\max\left\\{\frac{\pi}{2}h\left(a^{\prime}\right),\,\frac{\pi}{2}h\left(b^{\prime}\right),\,\frac{\pi}{2}\right\\}$
$\displaystyle=\mathcal{C}_{123}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}^{\prime}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\subset\mathcal{O}_{L}\\\
\mathfrak{p}\mid
c\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)^{2}h\left(b^{\prime}\right),$
(118)

where the line of reasoning about
$\max{\left\\{h\left(a^{\prime}\right),\,h\left(b^{\prime}\right),\,1,\,\frac{\pi}{d}\right\\}}$
follows from assumption (19). We are thus able to replace (3) with

$\max\left\\{h\left(\frac{u_{a}a^{\prime}}{u_{c}c^{\prime}}\right),\,h\left(\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)\right\\}<\mathcal{C}_{124}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}^{\prime}_{c}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\subset\mathcal{O}_{L}\\\
\mathfrak{p}\mid
c\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)^{2}h\left(b^{\prime}\right).$
(119)

As before, we now let
$S=S_{{}_{\infty}}\cup\left\\{\mathfrak{p}\,:\,\mathfrak{p}\mid
bc\mathcal{O}_{L}\right\\}$. Applying Lemma 8 again, following the same method
as above, in place of (3) we obtain that

$\max\left\\{h\left(-\frac{u_{a}}{u_{c}c^{\prime}}\right),\,h\left(-\frac{u_{b}b^{\prime}}{u_{c}c^{\prime}}\right)\right\\}<\mathcal{C}_{125}\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{q}\right)\left(\prod_{\begin{subarray}{c}\mathfrak{p}\subset\mathcal{O}_{L}\\\
\mathfrak{p}\mid
bc\mathcal{O}_{L}\end{subarray}}\log\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}\right)\right)^{2}h\left(a^{\prime}\right),$
(120)

where $\mathfrak{q}$ is the prime ideal of $\mathcal{O}_{L}$ of third largest
norm dividing $bc\mathcal{O}_{L}$. We note that this is not necessarily
$\mathfrak{p}_{b}$ or $\mathfrak{p}_{c}$, though it may have the same norm as
one of them, and could indeed be either of them.

We now follow the argument of the main text again, using the above
inequalities in place of (3) and (3) as necessary, and we end up obtaining

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{q}\right)\right)^{\frac{1}{3}}G^{\mathcal{C}_{126}\left(\frac{\log\log\log
G}{\log\log G}+\frac{1}{\log\log G}+\frac{\log\log G}{\log G}\right)}$ (121)

in place of (3). As before, $\frac{\log\log\log G}{\log\log G}$ is the
dominant term in the exponent of $G$, so we can write

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<\left(\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{q}\right)\right)^{\frac{1}{3}}G^{\mathcal{C}_{127}\frac{\log\log\log
G}{\log\log G}}.$ (122)

We explore some cases. If $\mathfrak{q}=\mathfrak{p}_{b}^{\prime}$ then

$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}^{\prime}\right)\leq
G.$

If $\mathfrak{q}=\mathfrak{p}_{c}^{\prime}$ then there exists a prime ideal
$\mathfrak{p}_{c}^{\prime\prime}$, the prime of second largest norm dividing
$c\mathcal{O}_{L}$. Note that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\leq\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)^{\prime\prime}\leq\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$,
and all these primes divide $abc\mathcal{O}_{L}$ so their norms contribute to
$G$. Thus, in this case we obtain that

$\displaystyle\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)$
$\displaystyle<\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime\prime}\right)$
$\displaystyle\leq G.$

We have dealt with the cases where $\mathfrak{q}=\mathfrak{p}_{b}^{\prime}$
and $\mathfrak{q}=\mathfrak{p}_{c}^{\prime}$. Using the notation above, there
are four further possibilities for $\mathfrak{q}$, namely
$\mathfrak{p}_{b},\,\mathfrak{p}_{c},\,\mathfrak{p}_{b}^{\prime\prime},\,\mathfrak{p}_{c}^{\prime\prime}$.
If $\mathfrak{q}=\mathfrak{p}_{b}^{\prime\prime}$ or
$\mathfrak{p}_{c}^{\prime\prime}$ then substituting into the above expression,
it follows from the definition of $G$ that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{q}\right)<G$.

On the other hand, if $\mathfrak{q}=\mathfrak{p}_{b}$ or $\mathfrak{p}_{c}$,
we can still upper bound this by $G$. Assume $\mathfrak{q}=\mathfrak{p}_{b}$.
Then we deduce that
$\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\leq\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\leq\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)$.
It follows then that

$\displaystyle\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)$
$\displaystyle<\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{a}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{b}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)\text{Nm}_{\mathbb{Q}}^{L}\left(\mathfrak{p}_{c}^{\prime}\right)$
$\displaystyle\leq G.$

The argument is symmetric so applies if $\mathfrak{q}=\mathfrak{p}_{b}$. Thus
in all cases we obtain that

$\displaystyle\log H_{L}(a,\,b,\,c)$
$\displaystyle<G^{\frac{1}{3}+\mathcal{C}_{128}\frac{\log\log\log G}{\log\log
G}}.$ (123)

We note that for given $a,\,b,\,c$, once we know the prime ideals dividing
$a\mathcal{O}_{K},\,b\mathcal{O}_{K}$ and $c\mathcal{O}_{K}$, the inequality
(122) may be stronger than that given in (123).

##  6 Some Remarks

A combination of methods and results by Györy and Yu [13] and Györy [10], [11]
with the method of Le Fourn [19] can be used directly to find results over the
base field, as done by Györy in [12]. Further, in terms of $S$, Györy improved
the $S$-unit bound given by Le Fourn. Györy’s result regarding the $abc$
conjecture is as follows.

Let $K$ be a number field and let $a,\,b,\,c:=a+b$ belong to $K^{*}$. Define

$N_{K}=\prod_{\upsilon}\textrm{Nm}_{\mathbb{Q}}^{K}\left(\mathfrak{p}\right)^{\text{ord}_{p}\left(\mathfrak{p}\right)},$

where $\upsilon$ is taken from the set of finite places such that
$\left|a\right|_{\upsilon},\,\left|b\right|_{\upsilon}$ and
$\left|c\right|_{\upsilon}$ are not all equal, and $p$ is the rational prime
such that $\mathfrak{p}\cap\mathbb{Z}=p$. Then for all $\varepsilon$ there is
a computable constant $\mathcal{C}_{129}$ depending only on
$d=\left[K:\mathbb{Q}\right],\,\Delta_{K}$ and $\varepsilon$ such that

$\log
H_{K}\left(a,\,b,\,c\right)<\mathcal{C}_{129}G_{K}^{\frac{1}{3}+\epsilon}.$

Györy’s combination of his method with that of Le Fourn’s enables him to state
his results entirely over the base field $K$. On the other hand, the
dependence on the norms of prime ideals in this paper allows us to state
corollaries depending on these norms, leading to the sub-exponential bound for
example. Further, we believe they may allow some attack at open problems such
as the smooth $abc$ conjecture [16].

For all these results, we have considered them in terms of $\log
H_{L}\left(a,\,b,\,c\right)$. We note that
$H_{L}\left(a,\,b,\,c\right)=H_{K}\left(a,\,b,\,c\right)^{h_{K}}$, as
$h_{K}=\left[L:K\right]$. Thus, as $h_{K}$ depends on the field, after taking
the logarithm we can incorporate the $h_{K}$ into our computable constant and
have the height in terms of the base field $K$. However, so far we have been
unable to do the same for the radical $G$.

##  7 Application to Effective Skolem-Mahler-Lech Problem

In this section we will use our main result to allow us to determine whether a
linear recurrence sequence of degree three with no repeated roots of the
characteristic polynomial has zeroes. As noted in the introduction, there
exists an algorithm to determine whether there are periodic zeroes, so we are
concerned with the case when there are only potentially finitely many zeroes.

###  7.1 Case Where all Terms are Coprime

Consider a linear recurrence sequence of the following form:

$a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+c_{3}a_{n-3},$

where the values of $a_{0},\,a_{1}$ and $a_{2}$ are known. We form the
characteristic polynomial of the sequence

$x^{3}-c_{1}x^{2}-c_{2}x-c_{3}$

and assume that this has distinct roots $r_{1},\,r_{2},\,r_{3}$. Let
$K=\mathbb{Q}\left(r_{1},\,r_{2},\,r_{3}\right)$. We further assume the roots
are pairwise coprime when considered as principal ideals of the ring of
integers $\mathcal{O}_{K}$.

By our assumptions, we know that we can write

$a_{n}=k_{1}r_{1}^{n}+k_{2}r_{2}^{n}+k_{3}r_{3}^{n}$

where $k_{1},\,k_{2},\,k_{3}$ are constants depending on $a_{0},\,a_{1}$ and
$a_{2}$. We further assume $k_{1},\,k_{2}$ and $k_{3}$ are coprime, but we
will look at ways to try and deal with this when not coprime later.

For ease, we use the result obtained by using Le Fourn’s Lemma, that is the
inequality given at (123). Assume there exists an $n$ such that $a_{n}=0$.
Explicitly,

$0=k_{1}r_{1}^{n}+k_{2}r_{2}^{n}+k_{3}r_{3}^{n}.$

We are in a position to use the result. Let $L=HCF(K)$ and define $G$ as
above. Then,

$\log
H\left(k_{1}r_{1}^{n},\,k_{2}r_{2}^{n},\,k_{3}r_{3}^{n}\right)<G^{\frac{1}{3}+\mathcal{C}_{128}\frac{\log\log\log
G}{\log\log G}}.$

Without loss of generality, assume that

$h\left(r_{1}\right)\leq h\left(r_{2}\right)\leq h\left(r_{3}\right).$

Note that

$H\left(k_{1}r_{1}^{n},\,k_{2}r_{2}^{n},\,k_{3}r_{3}^{n}\right)=H\left(\frac{k_{1}}{k_{3}}r_{1}^{n},\,\frac{k_{2}}{k_{3}}r_{2}^{n},\,r_{3}^{n}\right).$

Further, by comparing definitions,

$h\left(r_{3}^{n}\right)\leq\log
H\left(\frac{k_{1}}{k_{3}}r_{1}^{n},\,\frac{k_{2}}{k_{3}}r_{2}^{n},\,r_{3}^{n}\right).$

Moreover, $h\left(r_{3}^{n}\right)~{}=nh\left(r_{3}\right)$ [37]. Combining
all this we obtain that

$nh\left(r_{3}\right)<G^{\frac{1}{3}+\mathcal{C}_{128}\frac{\log\log\log
G}{\log\log G}}.$

It follows that

$n<\frac{G^{\frac{1}{3}+\mathcal{C}_{128}\frac{\log\log\log G}{\log\log
G}}}{h\left(r_{3}\right)},$

giving an upper bound for $n$.

Explicitly, given a recurrence relation satisfying the given conditions, we
first check whether there are any zeroes in arithmetic progressions. If so, we
are done. If not, we apply the above method, which gives an upper bound for
the maximal value of $n$ such that $a_{n}=0$. We numerically check the values
of $a_{x}$ for $x$ less than the obtained upper bound. This answers the
question as to whether the recurrence sequence has a zero.

######  Example 1.

Consider the linear recurrence sequence with
$a_{0}=31,\,a_{1}=112,\,a_{2}=452$ and

$a_{n}=10a_{n-1}-31a_{n-2}+30a_{n-3}.$

This sequence has characteristic polynomial

$x^{3}-10x^{2}+31x-30,$

with roots $2,\,3$ and $5$. Thus,

$a_{n}=k_{1}2^{n}+k_{2}3^{n}+k_{3}5^{n},$

where $k_{1},\,k_{2}$ and $k_{3}$ are to be found. They are found to be
$k_{1}=7,\,k_{2}=11$ and $k_{3}=13$, so

$a_{n}=7\cdot 2^{n}+11\cdot 3^{n}+13\cdot 5^{n}.$

This means $G=2\cdot 3\cdots 5\cdot 7\cdot 11\cdot 13=30030$. The largest
logarithmic height of the roots is $h\left(5\right)=\log 5$. It follows that
if $a_{n}=0$, then

$\displaystyle n$
$\displaystyle<\frac{30030^{\frac{1}{3}+\mathcal{C}_{128}\frac{\log\log\log
30030}{\log\log 30030}}}{\log 5}$ $\displaystyle<20\cdot
43^{\mathcal{C}_{130}}.$

In principle, $\mathcal{C}_{128}$ and $\mathcal{C}_{130}$ can be computed
following the proof given in this paper. This gives an upper bound for $n$.

In this example, once we derive $a_{n}=7\cdot 2^{n}+11\cdot 3^{n}+13\cdot
5^{n}$, it is clear there are no zeroes. With more complicated examples, it
may not be so obvious.

###  7.2 Case Where Terms are Not Coprime

Assume we have a linear recurrence relation as above with characteristic
polynomial $f(x)$ with roots $r_{1},\,r_{2},\,r_{3}$. Assume there are
constants $k_{1},\,k_{2},\,k_{3}$ so

$a_{n}=k_{1}r_{1}^{n}+k_{2}r_{2}^{n}+k_{3}r_{3}^{n}.$

We assume nothing about coprimeness. If they are all coprime, we’re done as
above. We thus assume $k_{1}r_{1}^{n},\,k_{2}r_{2}^{n},\,k_{3}r_{3}^{n}$ are
not coprime.

If there exists an $n$ such that $a_{n}=0$, then the same prime ideal must
divide all 3 terms. We can see this as if

$0=k_{1}r_{1}^{n}+k_{2}r_{2}^{n}+k_{3}r_{3}^{n},$

then

$-k_{1}r_{1}^{n}=k_{2}r_{2}^{n}+k_{3}r_{3}^{n},$

and it follows if a prime ideal divides two of these as ideals, it has to
divide the third.

More rigorously,

$\text{ord}_{\mathfrak{p}}\left(a+b\right)\geq\min\left\\{\text{ord}_{\mathfrak{p}}\left(a\right),\,\text{ord}_{\mathfrak{p}}\left(b\right)\right\\},$
(124)

with equality when
$\text{ord}_{\mathfrak{p}}\left(a\right)\neq\text{ord}_{\mathfrak{p}}\left(b\right)$.
The claim directly follows from this. It also follows from this that at least
two of the terms are divisible by the prime ideal to the same order.

We consider these as ideals of $\mathcal{O}_{K}$. Assume that $\mathfrak{q}$
is a prime ideal of $\mathcal{O}_{K}$ dividing all three terms, and that
$\text{ord}_{\mathfrak{q}}\left(k_{1}r_{1}^{n}\right)=\text{ord}_{\mathfrak{q}}\left(k_{2}r_{2}^{n}\right)=l$.
We write

$\displaystyle k_{1}r_{1}^{n}\mathcal{O}_{K}$
$\displaystyle=\mathfrak{p}_{k_{1},\,1}^{e_{k_{1},\,1}}\cdots\mathfrak{p}_{k_{1},\,a}^{e_{k_{1},\,a}}\mathfrak{p}_{r_{1},\,1}^{n\cdot
e_{r_{1},\,1}}\cdots\mathfrak{p}_{r_{1},\,b}^{n\cdot
e_{r_{1},\,b}}\mathfrak{q}^{l}$ $\displaystyle k_{2}r_{2}^{n}\mathcal{O}_{K}$
$\displaystyle=\mathfrak{p}_{k_{2},\,1}^{e_{k_{2},\,1}}\cdots\mathfrak{p}_{k_{2},\,a}^{e_{k_{2},\,c}}\mathfrak{p}_{r_{2},\,1}^{n\cdot
e_{r_{2},\,1}}\cdots\mathfrak{p}_{r_{2},\,b}^{n\cdot
e_{r_{2},\,d}}\mathfrak{q}^{l}$ $\displaystyle k_{3}r_{3}^{n}\mathcal{O}_{K}$
$\displaystyle=\mathfrak{p}_{k_{3},\,1}^{e_{k_{3},\,1}}\cdots\mathfrak{p}_{k_{3},\,f}^{e_{k_{3},\,f}}\mathfrak{p}_{r_{3},\,1}^{n\cdot
e_{r_{3},\,1}}\cdots\mathfrak{p}_{r_{3},\,b}^{n\cdot
e_{r_{3},\,g}}\mathfrak{q}^{m},$

where these ideals are prime ideals of $\mathcal{O}_{K}$.

Note, it may be the case that $l=m$. Also, if $\mathfrak{q}\mid r_{1}^{n}$,
this implies that $l=an$ for some $a$, but we will see this doesn’t matter for
the argument. Finally, it may be that there is a further prime ideal that
divides all three terms;p if so, we apply the following process iteratively on
all prime ideals dividing all three terms.

We now move the the Hilbert Class Field $L$. All the ideals above are
principal as ideals of $\mathcal{O}_{L}$, so we can write

$\displaystyle k_{1}r_{1}^{n}$
$\displaystyle=u_{1}{p}_{k_{1},\,1}^{e_{k_{1},\,1}}\cdots{p}_{k_{1},\,a}^{e_{k_{1},\,a}}{p}_{r_{1},\,1}^{n\cdot
e_{r_{1},\,1}}\cdots{p}_{r_{1},\,b}^{n\cdot e_{r_{1},\,b}}{q}^{l}$
$\displaystyle k_{2}r_{2}^{n}$
$\displaystyle=u_{2}{p}_{k_{2},\,1}^{e_{k_{2},\,1}}\cdots{p}_{k_{2},\,a}^{e_{k_{2},\,c}}{p}_{r_{2},\,1}^{n\cdot
e_{r_{2},\,1}}\cdots{p}_{r_{2},\,b}^{n\cdot e_{r_{2},\,d}}{q}^{l}$
$\displaystyle k_{3}r_{3}^{n}$
$\displaystyle=u_{3}{p}_{k_{3},\,1}^{e_{k_{3},\,1}}\cdots{p}_{k_{3},\,f}^{e_{k_{3},\,f}}{p}_{r_{3},\,1}^{n\cdot
e_{r_{3},\,1}}\cdots{p}_{r_{3},\,b}^{n\cdot e_{r_{3},\,g}}{q}^{m},$

where the terms on the right hand side are all elements of $L$ that generate
the relevant principal ideals. Thus, we can now write

$\displaystyle a_{n}=$ $\displaystyle
u_{1}{p}_{k_{1},\,1}^{e_{k_{1},\,1}}\cdots{p}_{k_{1},\,a}^{e_{k_{1},\,a}}{p}_{r_{1},\,1}^{n\cdot
e_{r_{1},\,1}}\cdots{p}_{r_{1},\,b}^{n\cdot e_{r_{1},\,b}}{q}^{l}$
$\displaystyle+u_{2}{p}_{k_{2},\,1}^{e_{k_{2},\,1}}\cdots{p}_{k_{2},\,a}^{e_{k_{2},\,c}}{p}_{r_{2},\,1}^{n\cdot
e_{r_{2},\,1}}\cdots{p}_{r_{2},\,b}^{n\cdot e_{r_{2},\,d}}{q}^{l}$
$\displaystyle+u_{3}{p}_{k_{3},\,1}^{e_{k_{3},\,1}}\cdots{p}_{k_{3},\,f}^{e_{k_{3},\,f}}{p}_{r_{3},\,1}^{n\cdot
e_{r_{3},\,1}}\cdots{p}_{r_{3},\,b}^{n\cdot e_{r_{3},\,g}}{q}^{m}.$

By (124), we see that $m\geq l$. We thus divide through the above equation by
$q^{l}$. After repeating this for all prime ideals dividing all three terms,
the remaining terms on the right hand side will all be coprime. We now assume
that there exists an $n$ such that $a_{n}=0$, and we are in the same position
as section 8.1, and the argument follows identically.

######  Remarks.

We note that this application also follows from Györy’s result [12].

##  8 Smooth Solutions to the $abc$ Conjecture

In this section we will prove Theorem 5, as given in the introduction. We will
first prove the following lemma.

######  Lemma 9.

Let $\left(X,\,Y,\,Z\right)\in\mathbb{Z}^{3}$ be a triple with smoothness
$S\left(X,\,Y,\,Z\right)$ and radical $G\left(X,\,Y,\,Z\right)$ defined as
above. Then

$G\left(X,\,Y,\,Z\right)\leq e^{3S\left(X,\,Y,\,Z\right)}.$ (125)

###### Proof.

From [30], we know that for $n\geq 6$,

$\frac{p_{n}}{n}<\log n+\log\log n,$

where $p_{n}$ denotes the n’th rational prime.

It follows that for $n\geq 6$,

$p_{n}\leq 2n\log n.$

Indeed, computationally we can check primes $2,\,3,\,5,\,7,\,11$ and we find
that for $n>2$ the above inequality holds.

Further, by Rosser’s Theorem [29],

$p_{n}>n\log n.$

Thus, it follows that the product of the first $k$ primes satisfies the
following inequality:

$\displaystyle\prod_{i=1}^{k}p_{i}$ $\displaystyle\leq 2\cdot
3\cdot\prod_{i=3}^{k}2i\log i$ $\displaystyle=2\cdot 3\cdot
2^{k-2}\left(4\cdot 5\cdots k\right)\cdot\prod_{i=3}^{k}\log i$
$\displaystyle=2^{k-2}\cdot k!\cdot\prod_{i=3}^{k}\log i$ $\displaystyle\leq
2^{k}\cdot k^{k}\cdot\prod_{i=3}^{k}\log i.$ (126)

We now take logs of each side of the inequality attaining

$\displaystyle\log\left(\prod_{i=1}^{k}p_{i}\right)$
$\displaystyle\leq\log\left(2^{k}\cdot k^{k}\cdot\prod_{i=3}^{k}\log i\right)$
$\displaystyle=k\log 2+k\log k+\log\left(\prod_{i=3}^{k}\log i\right)$
$\displaystyle=k\log 2+k\log k+\sum_{i=3}^{k}\log\log i$ $\displaystyle\leq
k\log 2+k\log k+k\log\log k$ $\displaystyle\leq k\log 2+k\log k+k\log k$
$\displaystyle\leq 3k\log k$ $\displaystyle\leq 3p_{k}$ (127)

where the last line follows from Rosser’s Theorem.

It thus follows that for a triple of pairwise coprime integers satisfying
$X+Y=Z$ with smoothness $S\left(X,\,Y,\,Z\right)$,

$\displaystyle G\left(X,\,Y,\,Z\right)$
$\displaystyle\leq\prod_{\begin{subarray}{c}p\textrm{ prime}\\\ p\leq
S\left(X,\,Y,\,Z\right)\end{subarray}}p$ $\displaystyle\leq
e^{3S\left(X,\,Y,\,Z\right)},$ (128)

by the above inequality. ∎

We are now in a position to prove Theorem 5.

###### Proof.

By Northcott’s Theorem, we can assume in the following that
$H\left(X,\,Y,\,Z\right)>B$ for any given bound $B$ as we will only be
excluding finitely many possible solutions $\left(X,\,Y,\,Z\right)$ satisfying
(8).

For ease of notation, write

$T:=\log\log H\left(X,\,Y,\,Z\right).$ (129)

By assumption, we have that

$S\left(X,\,Y,\,Z\right)<T\frac{\log T}{\log\log T\phi\left(T\right)}.$ (130)

We study triples $\left(X,\,Y,\,Z\right)$ satisfying (130), and note that for
a sufficiently large $H\left(X,\,Y,\,Z\right)$,

$T\frac{\log T}{\log\log T\phi\left(T\right)}<\left(\log
H\left(X,\,Y,\,Z\right)\right)^{\frac{1}{2}},$

so we can apply Corollary 10. We note the choice of the exponent to be
$\frac{1}{2}$ is incidental; indeed any exponent less than $\frac{2}{3}$ could
have been chosen. Further, the Hilbert Class Field of $\mathbb{Q}$ is itself
$\mathbb{Q}$ [7], so the radical $G$ in this case is defined over $\mathbb{Q}$
and coincides with the radical given in [16].

By Corollary 10, we know that

$\log H\left(X,\,Y,Z\right)<G^{\mathcal{C}_{89}\frac{\log\log\log G}{\log\log
G}}.$

From the upper bound for $G$ given at (125) in Lemma 9, we obtain from the
above that

$\displaystyle\log H\left(X,\,Y,\,Z\right)$
$\displaystyle<e^{3\mathcal{C}_{89}S\left(X,\,Y,\,Z\right)\frac{\log\log\log
e^{3S\left(X,\,Y,\,Z\right)}}{\log\log e^{3S\left(X,\,Y,\,Z\right)}}}$
$\displaystyle=e^{\mathcal{C}_{131}S\left(X,\,Y,\,Z\right)\frac{\log\log
3S\left(X,\,Y,\,Z\right)}{\log 3S\left(X,\,Y,\,Z\right)}}.$ (131)

It follows from the above that

$\log\log
H\left(X,\,Y,\,Z\right)=T<\mathcal{C}_{131}S\left(X,\,Y,\,Z\right)\frac{\log\log
3S\left(X,\,Y,\,Z\right)}{\log 3S\left(X,\,Y,\,Z\right)}.$ (132)

We recall that by Northcott’s Theorem again, we can assume that
$S\left(X,\,Y,\,Z\right)$ can be larger than any given constant while only
dropping finitely many solutions to $X+Y=Z$. Thus, only loosing finitely many
solutions, for sufficiently large $S\left(X,\,Y,\,Z\right)$ it follows from
(132) that

$T<\mathcal{C}_{132}S\left(X,\,Y,\,Z\right)\frac{\log\log
S\left(X,\,Y,\,Z\right)}{\log S\left(X,\,Y,\,Z\right)}.$ (133)

Taking logarithms on both side of inequality (133), we can assume
$S\left(X,\,Y,\,Z\right)$ is large enough to give that

$\displaystyle\log T$
$\displaystyle<\log\left(\mathcal{C}_{132}S\left(X,\,Y,\,Z\right)\frac{\log\log
S\left(X,\,Y,\,Z\right)}{\log S\left(X,\,Y,\,Z\right)}\right)$
$\displaystyle=\log\mathcal{C}_{132}+\log S\left(X,\,Y,\,Z\right)+\log\log\log
S\left(X,\,Y,\,Z\right)-\log\log S\left(X,\,Y,\,Z\right)$ $\displaystyle<2\log
S\left(X,\,Y,\,Z\right).$ (134)

For ease later, we divide the above by $2$ to give that

$\frac{1}{2}\log T<\log S\left(X,\,Y,\,Z\right).$ (135)

For triples $\left(X,\,Y,\,Z\right)$ satisfying (130), taking logarithms in
inequality (130) we deduce that

$\displaystyle\log S\left(X,\,Y,\,Z\right)$
$\displaystyle<\log\left(T\frac{\log T}{\log\log T\phi\left(T\right)}\right)$
$\displaystyle=\log T+\log\log T-\log\log\log T-\log\phi\left(T\right)$
$\displaystyle<2\log T.$ (136)

We note that this also implies that for sufficiently large
$S\left(X,\,Y,\,Z\right)$,

$\log\log S\left(X,\,Y,\,Z\right)<2\log\log T.$

Substituting this and (135) into (130) we obtain that

$\displaystyle S\left(X,\,Y,\,Z\right)$ $\displaystyle<T\frac{\log T}{\log\log
T\phi\left(T\right)}$ $\displaystyle<T\frac{2\log
S\left(X,\,Y,\,Z\right)}{\frac{1}{2}\log\log\left(S\left(X,\,Y,\,Z\right)\right)\phi\left(T\right)}$
$\displaystyle=4T\frac{\log
S\left(X,\,Y,\,Z\right)}{\log\log\left(S\left(X,\,Y,\,Z\right)\right)\phi\left(T\right)}.$
(137)

Rearranging the above we obtain that

$\frac{1}{4}\frac{S\left(X,\,Y,\,Z\right)\log\log
S\left(X,\,Y,\,Z\right)}{\log S\left(X,\,Y,\,Z\right)}\phi\left(T\right)<T$
(138)

We now have two inequalities relating $S$ and $T$, namely (133) and (138)
given above. We compare these directly to find that

$\displaystyle\frac{1}{4}\frac{S\left(X,\,Y,\,Z\right)\log\log
S\left(X,\,Y,\,Z\right)}{\log
S\left(X,\,Y,\,Z\right)}\phi\left(T\right)<T<\mathcal{C}_{132}\frac{S\left(X,\,Y,\,Z\right)\log\log
S\left(X,\,Y,\,Z\right)}{\log S\left(X,\,Y,\,Z\right)},$ (139)

which we rewrite as

$\displaystyle\frac{S\left(X,\,Y,\,Z\right)\log\log
S\left(X,\,Y,\,Z\right)}{\log
S\left(X,\,Y,\,Z\right)}\phi\left(T\right)<T<\mathcal{C}_{133}\frac{S\left(X,\,Y,\,Z\right)\log\log
S\left(X,\,Y,\,Z\right)}{\log S\left(X,\,Y,\,Z\right)}.$ (140)

Cancelling terms on both sides gives us that