# On solutions of the Diophantine equation $\mathcal{P}_{m}-L_{n}=c$

Pagdame TIEBEKABE∗,1,2, Serge ADONSOU3 $\&$ Ismaïla DIOUF1

###### Abstract

In this article, we determine all the integers $c$ having at least two
representations as difference between two linear recurrent sequences. This is
a variant of the Pillai’s equation. This equation is an exponential
Diophantine equation. The proof of our main theorem uses lower bounds for
linear forms of logarithms, properties of continued fractions, and a version
of the Baker-Davenport reduction method in Diophantine approximation.

Keywords: Linear forms in logarithms; Diophantine equations; Pillai’s problem;
Linear recurrent sequences.
2020 Mathematics Subject Classification: 11B39, 11J86, 11D61.

  * *

: Corresponding author

  * 1

:Université Cheikh Anta Diop (UCAD), Laboratoire d’Algèbre, de Cryptologie, de
Géométrie Algébrique et Applications (LACGAA), Dakar, Sénégal

  * 2

: Université de Kara, Kara-Togo

  * 3

: African Insitute for Mathematical Sciences (AIMS), Afrique du Sud

##  1 Introduction

It is well known that the sequence $\\{\mathcal{P}_{k}\\}_{k\geq 1}$ of
Padovan numbers is defined by

$\mathcal{P}_{0}=\mathcal{P}_{1}=\mathcal{P}_{2}=1,\quad\mathcal{P}_{k+3}=\mathcal{P}_{k+1}+\mathcal{P}_{k},\quad
k\geq 0.$

The first Padovan numbers are

$1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265\ldots$

The sequence $\\{L_{k}\\}_{k\geq 1}$ of Lucas numbers is defined by

$L_{0}=2,\quad L_{1}=1,\quad L_{k+2}=L_{k+1}+L_{k},\quad k\geq 0.$

The first Lucas numbers are

$2,1,3,4,7,11,18,29,47,76,123,\ldots$

In this article, we are interested in the determination of solutions of the
Diophantine equation

$\mathcal{P}_{m}-L_{n}=c$ (1)

for fixed $c$ and $m$, $n$ the unknowns. In particular, we are interested in
integers $c$ admitting at least two representations as the difference between
a Padovan number and a Lucas number. It is a variant of the equation

$a^{x}-b^{y}=c,$ (2)

in positive integers $(x,y)$ where $a,b,c$ are fixed positive integers. The
history of the equation (2) is very rich and goes back to $1935$. Subbayya
Sivasankaranarayana Pillai ($1901-1950$) is an Indian mathematician
specializing in number theory. He has written several articles on perfect
powers. A perfect power is a positive integer of the form $a^{x}$ where $a\geq
1$ and $x\geq 1$ are natural integers. In $1931$, S.S. Pillai proved in [25]
that for all positive integers $a$ and $b$ fixed, both, the number of
solutions $(x,y)$ of the Diophantine inequalities $0<a^{x}-b^{y}\leq c$ is
asymptotically equal to

$\dfrac{(\log c)^{2}}{2(\log a)(\log b)}.$ (3)

when $c$ tends to infinity. It is very interesting to read the $62$ page of
this article to see how this result was obtained. This result follows from the
attempt to prove that the equation

$m^{x}-n^{y}=a.$

has only a finite number of integral solutions. In this equation, $m$, $n$ and
$a$ are fixed. The unknowns are $x$ and $y$. After several years, he repeated
this same equation, but this time with $m$, $n$, $x$ and $y$ as unknowns,
fixing only $a$. Research on this equation began with S.S. Pillai in $1931$.
In $1936$, A. Herschfeld ([23] and [24]) continued the research and showed
that if $|c|$ is a large enough integer, then the equation

$2^{x}-3^{y}=c$ (4)

has at most one solution $(x,y)$ with $x$ and $y$ being positive integers.

This result is no longer true for $|c|$ small enough. By classical methods,
Herschfeld demonstrated that only triples of integers $(x,y,c)$ with positive
$x$ and $y$ such that $2^{x}-3^{y}=c$ is given for $|c|\leq 10$ by:

$(2,1,1),(1,1,-1),(3,2,-1),(3,1,5),(5,3,5),(2,2,5),(4,2,7),(1,2,-7).$

So if $x>5$ or $y>3$, then $|2^{x}-3^{y}|>10$. Proceeding in the same way, he
proved that if $x>8$ or $y>5$, then $|2^{x}-3^{y}|>100$.

S.S. Pillai ([23] and [24]) extended Herschfeld’s results to the more general
case of exponential Diophantine equations

$a^{x}-b^{y}=c,$ (5)

where $a$ , $b$ and $c$ are nonzero integers fixed with gcd$(a,b)=1$ and
$a>b\geq 2$. He showed that there exists a positive integer $c_{0}(a,b)$ such
that, for $|c|>c_{0}(a,b)$, this equation has at most one solution. This proof
does not give the explicit value of $c_{0}(a,b)$. In the special case of the
Herschfeld equation with $(a,b)=(2,3)$, S.S. Pillai conjectured that
$c_{0}(a,b)=13$ and said that the integer $c$ which has two representations of
the form $3^{n}-2^{m}$ are the elements of the set $\\{-13,-5,1\\}$. This
conjecture was solved by R. J. Stroeker and R. Tijdeman in $1982$ by measuring
the linear independence of the forms of logarithms of algebraic numbers.

######  Conjecture 1.1 (Conjecture de Pillai).

For any integer $k\geq 1$, the Diophantine equation

$x^{n}-y^{m}=k$ (6)

admits a finite number of positive integer solutions $(n,m,x,y)$, with $n\geq
2$ and $m\geq 2$.

Since then, several variants of the equation (5) have been intensively
studied. Recent results related to the equation $H_{n}-G_{n}=c$ where
$(H_{n})_{n\geq 0}$ and $(G_{n})_{n\geq 0}$ represent linear recurrent
sequences are obtained by M Ddamulira et al in which they solved this type of
Pillai equations with Fibonacci numbers and powers of $2$ (see [13]), M.
Ddamulira et al solved the case with generalized Fibonacci numbers and powers
of $2$ (see [14]), and Bravo et al solved the case of Tribonacci numbers and
powers of $2$ (see [10]). We have also solved the case of Padovan numbers and
Lucas numbers, by determining the numbers $c$ which have at least two
representations as difference of Padovan and Lucas numbers. More simply, we
solved the equation $\mathcal{P}_{m}-L_{n}=c$ with $m>3$. The articles [17,
18, 19, 20, 21, 22] also discuss variants of the Pillai equation and other
Diophantine equations solved by the method of logarithmic linear forms. The
purpose of this article is to prove the following result.

######  Théorème 1.1.

The only integers $c$ having at least two representations of the form
$\mathcal{P}_{m}-L_{n}$ are

$\begin{array}[]{c}c\in\\{-643,-310,-171,-74,-48,-27,-26,-13,-11,-9,-8,-6,-4,-2,-1,0,1,2,3,\\\
4,5,6,8,9,10,14,17,18,19,20,26,36,38,47,64,68,75,85,189,2864,58269\\}\end{array}$

We organize this article as follows. In the next section, we recall some
useful results for the proof of the theorem 1.1. The proof of the theorem 1.1
is done in the last section.

##  2 Auxiliary results

###  2.1 Some properties of Lucas and Padovan sequences

We recall here some properties of Lucas $\\{L_{k}\\}$ and Padovan
$\\{\mathcal{P}_{k}\\}_{k\geq 0}$ sequences which are useful to prove our
theorem.
The characteristic equation is

$x^{3}-x-1=0,$

has roots $\alpha,\beta,\gamma=\overline{\beta}$, where

$\alpha=\dfrac{r_{1}+r_{2}}{6},\quad\beta=\dfrac{-r_{1}-r_{2}+i\sqrt{3}(r_{1}-r_{2})}{12},$

and

$r_{1}=\sqrt[3]{108+12\sqrt{69}}\;\text{ and }\
;r_{2}=\sqrt[3]{108-12\sqrt{69}}.$

Cardan’s formulas give for the real root the plastic number or silver number:

$\sqrt[3]{\dfrac{1}{2}+\dfrac{\sqrt{69}}{18}}+\sqrt[3]{\dfrac{1}{2}-\dfrac{\sqrt{69}}{18}}\approx
1,32472.$

Also, Binet’s formula is

$\mathcal{P}_{k}=a\alpha^{k}+b\beta^{k}+c\gamma^{k},\ ;\text{ for all }k\geq
0,$ (7)

where

$\begin{array}[]{lll}a&=&\dfrac{(1-\beta)(1-\gamma)}{(\alpha-\beta)(\alpha-\gamma)}=\dfrac{1+\alpha}{-\alpha^{2}+3\alpha+1},\vspace{1mm}\\\
b&=&\dfrac{(1-\alpha)(1-\gamma)}{(\beta-\alpha)(\beta-\gamma)}=\dfrac{1+\beta}{-\beta^{2}+3\beta+1},\vspace{1mm}\\\
c&=&\dfrac{(1-\alpha)(1-\beta)}{(\gamma-\alpha)(\gamma-\beta)}=\dfrac{1+\gamma}{-\gamma^{2}+3\gamma+1}=\overline{b}.\end{array}$
(8)

Numerically, we have

$\begin{array}[]{l}1.32<\alpha<1.33,\\\
0.86<|\beta|=|\gamma|=\alpha^{-1/2}<0.87,\\\ 0.72<a<0.73,\\\
0.24<|b|=|c|<0.25.\end{array}$ (9)

Using induction, we can show that

$\alpha^{k-2}\leq\mathcal{P}_{k}\leq\alpha^{k-1},$ (10)

for all $k\geq 4$.

On the other hand, let $(\delta,\eta)=((1+\sqrt{5})/2,(1-\sqrt{5})/2)$ be the
roots of the characteristic equation $x^{2}-x-1=0$ of the Lucas sequence
$\\{L_{k}\\}_{k\geq 0}$. Binet’s formula for $L_{k}$

$L_{k}=\delta^{k}+\eta^{k}\quad\text{is valid for all }k\geq 0.$ (11)

This easily implies that the inequality

$\delta^{k-1}\leq L_{k}\leq\delta^{k+1}$ (12)

holds for all positive integers $k.$

Now let’s discuss the notions of naive height and absolute logarithmic height.

####  2.1.1 Algebraic height

In this section, we will introduce the notion of algebraic height which is
very useful as we will see later. We begin by defining the naive height and
then deducing from it the absolute logarithmic height.

######  Définition 2.1 (Naive height).

For any algebraic number $\gamma$, we define the height of $\gamma$ by:

$\displaystyle H(\gamma)=\max(|a_{d}|,\ldots,|a_{0}|),$

where $f(x)=a_{d}x^{d}+\cdots+a_{1}x+a_{0}$ is a minimal polynomial of
$\gamma$ over $\mathbb{Z}$. $H(\gamma)$ is called the naive height of
$\gamma$.

######  Exemple 2.1.

Let $\alpha$ be an algebraic number:

  * •

Si $\gamma\in\mathbb{Z}$, $H(\gamma)=|\gamma|$.

  * •

Si $\gamma\in\mathbb{Q}\left(\text{i.e.}\ \gamma=\dfrac{b}{a}\text{ avec
\text{pgcd}}(a,b)=1\right)$, $H(\gamma)=\max\\{|a|,|b|\\}$,

For any algebraic number $\gamma$, we have the following identity:

$H(\gamma)=|a_{d}|\prod\limits_{i=1}^{d}\max\\{1,|\gamma_{i}|\\},$ (13)

where $\gamma_{i}$ represent the roots of the minimal polynomial and
$f(x)=a_{d}\prod\limits_{i=1}^{d}(x-\gamma^{(i)})$ is the minimal polynomial
of $\ gamma$. We define in the next subsection, another height deduced from
the previous one called absolute logarithmic height. It is the most used.

######  Définition 2.2 (Absolute logarithmic height).

For a nonzero algebraic number of degree $d$ over $\mathbb{Q}$ where the
minimal polynomial over $\mathbb{Z}$ is
$f(x)=a_{d}\prod\limits_{i=1}^{d}(x-\gamma^{(i)})$, we denote by:

$h(\gamma)=\dfrac{1}{d}\left(\log|a_{d}|+\sum\limits_{i=1}^{d}\log\max\\{1,|\gamma^{i}|\\}\right)=\dfrac{1}{d}\log\mathrm{M}(\gamma),$
(14)

the usual absolute logarithmic height of $\gamma$.

The properties of absolute logarithmic height are as follows:

######  Proposition 2.1 (Y. F. Bilu, Y. Bugeaud et M. Mignotte).

  1. 1.

Let $\gamma$, $\delta$ be two nonzero algebraic numbers. We have

     * •

$h(\gamma\delta)\leq h(\gamma)+h(\delta)$,

     * •

$h(\gamma+\delta)\leq h(\gamma)+h(\delta)+\log 2$.

  2. 2.

For any algebraic number $\gamma$ and $n\in\mathbb{Z}$ (with $\gamma\neq 0$
and if $n<0$) we have:
$h(\gamma^{n})=|n|h(\gamma)$.

More generally, for $\gamma_{1},\gamma_{2},\cdots,\gamma_{n}$, $n$ algebraic
numbers, we have:

  * •

$h(\gamma_{1}\gamma_{2}\cdots\gamma_{n})\leq
h(\gamma_{1})+h(\gamma_{2})+\cdots+h(\gamma_{n})$

  * •

$h(\gamma_{1}+\gamma_{2}+\cdots+\gamma_{n})\leq
h(\gamma_{1})+h(\gamma_{2})+\cdots+h(\gamma_{n})+\log n$.

######  Exemple 2.2.

Let $\gamma$ be an algebraic number

  1. 1.

If $\gamma$ is the root of $x^{2}-2x-1$, then
$h(\gamma)=\frac{1}{2}(\log\max\\{1,|\alpha|\\}+\log\max\\{1,|\beta|\\})$,
$\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$, then
$h(\gamma)=\frac{1}{2}\log\alpha$.

  2. 2.

Let $h(\gamma)$ be determined with $\gamma=\dfrac{\alpha^{n}-1}{2\sqrt{2}}$
where $\alpha=1+\sqrt{2}$. From propositions (9.1) and (9.2), we know that
$2\sqrt{2}P_{n}=\alpha^{n}-\beta^{n}$ and $Q_{n}=\alpha^{n}+\beta^{n}$.

So,

$\begin{array}[]{lllll}&4\sqrt{2}\gamma+2&=&2\sqrt{2}P_{n}+Q_{n}\\\
&4\sqrt{2}\gamma-2\sqrt{2}P_{n}&=&Q_{n}-2\\\
&8(4\gamma^{2}-4P_{n}\gamma+P_{n}^{2})&=&Q_{n}^{2}-4Q_{n}+4\\\
&8(4\gamma^{2}-4P_{n}\gamma)&=&Q_{n}^{2}-8P_{n}^{2}-4Q_{n}+4\\\
&8(4\gamma^{2}-4P_{n}\gamma)&=&4(-1)^{n}-4Q_{n}+4.\end{array}$

We obtain the minimal polynomial of $\gamma$ divides
$8x^{2}-8P_{n}x-((-1)^{n}+1-Q_{n})$. Which implies $a(x-\gamma)(x-1+\gamma)$
with $a\in\\{1,2,4,8\\}$.

$\displaystyle h(\gamma)=\dfrac{1}{a}\left(\log
a+\log\max\\{1,|\gamma|\\}+\log\max\\{1,|1-\gamma|\\}\right).$

Let us now state the theorems of Stewart, Baker and Wüstholz, before that of
Matveev.

######  Théorème 2.1 (1993, Baker and Wustholz).

If $\Lambda\neq 0$, then

$|\Lambda|>\exp\left(-(16nd)^{2n+4}.\log A_{1}...\log A_{n}.\log B\right)$
(15)

with $A_{i}=\max\\{H(\alpha_{i}),e\\}$, for $i=1,\cdots,n$;
$B=\max\\{|b_{1}|,\cdots,|b_{n}|,e\\}$ and
$d=[\mathbb{Q}(\alpha_{1},\cdots,\alpha_{n}):\mathbb{Q}]$.

######  Théorème 2.2 (A. Baker and G. Wüstholz).

Let $K$ be the field of algebraic numbers generated by
$\alpha_{1},\cdots,\alpha_{n}$ of degree $d$ over $\mathbb{Q}$. Let
$\alpha_{1},\cdots,\alpha_{n}$ $\in K^{*}$ and $b_{1},\cdots,b_{n}$
$\in\mathbb{Z^{*}}$.

Suppose $B^{*}=\max\\{|b_{1}|,\cdots,|b_{n}|\\}$ and $w=A_{1}A_{2}...A_{n}$
with
$|\Lambda|\geq\dfrac{1}{d}\left(\max\\{h(\alpha_{j}),|\log(\alpha_{j})|,1\\}\right)$
$(1\leq j\leq n)$.

Suppose also that $\Gamma\neq 0$, then;

$\log(|\Gamma|)>-18(n+1)!n^{n+1}(32d)^{n+2}w\log(2nd)\log(B^{*})$.

Let us now state the result of E. Matveev [6] which is the most used to solve
certain Diophantine equations.

######  Théorème 2.3 (E. M. Matveev).

Let $K$ be an algebraic field of numbers of degree $d$ over $\mathbb{Q}$. If
$K\subset\mathbb{R}$, set $\xi=1$ otherwise $\xi=2$. Let
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{K}^{*}$ and
$b_{1},\ldots,b_{n}\in\mathbb{Z}^{*}$. Suppose

$B^{*}=\max\\{|b_{1}|,\ldots,|b_{n}|\\},\ w=A_{1}A_{2}\ldots A_{n},\
A_{j}\geq\max\\{dh(\alpha_{j}),|\log(\alpha_{j}),0.16|\\}$

with ($1\leq j\leq n$) and

$\Gamma=b_{1}\log(\alpha_{1})+\cdots+b_{n}\log(\alpha_{n}).$

If $\Gamma\neq 0$, then

$\log(|\Gamma|)>-C_{1}(n)d^{2}w\log(\mathrm{e}d)\log(\mathrm{e}B^{*})$

with

$\
C_{1}(n)>\min\\{\frac{1}{\xi}(0.5\mathrm{e}n)^{\xi}30^{n+3}n^{3.5},2^{6n+20}\\}.$

More simply, Y. Bugeaud, M. Mignotte and S. Siksek established the following
result.

######  Théorème 2.4 (Y. Bugeaud, M. Mignotte, and S. Siksek).

Let $n\geq 1$ be an integer. Let $K$ be the field of algebraic numbers of
degree $d$. Let $\alpha_{1},...,\alpha_{n}$ be nonzero elements of $K$ and let
$b_{1},b_{2},...,b_{n}$ be integers,

$B=\max\\{|b_{1}|,...,|b_{n}|\\}$,

and

$\Lambda=\alpha_{1}^{b_{1}}...\alpha_{n}^{b_{n}}-1$.

Let $A_{1},...,A_{n}$ be real numbers such that

$A_{j}\geq\max\\{dh(\alpha_{j}),|\log(\alpha_{j}),0.16|\\}$, $1\leq j\leq n$.

Assuming $\Lambda\neq 0$, we have:

$\log|\Lambda|>-3\times 30^{n+4}\times(n+1)^{5.5}\times d^{2}\times
A_{1}...A_{n}(1+\log d)(1+\log nB).$

If $K$ is real, then

$\log|\Lambda|>-1.4\times 30^{n+3}\times(n)^{4.5}\times d^{2}\times
A_{1}...A_{n}(1+\log d)(1+\log B).$

Note that for some values of $n$, the lower bound of the logarithm proposed by
E.M. Matveev is better (slightly) than that of Baker and Wüstholz.

When $n=2$ and $\alpha_{1},\alpha_{2}$ multiplicatively independent, we have
these few results obtained by Laurent, Mignotte, Nesterenko ( [15], Corollary
$2$, pp. 288).

Let in this case $B_{1}$, $B_{2}$ be real numbers greater than 1 such that:

$\log
B_{i}\geq\max\left\\{h(\alpha_{i}),\dfrac{|\log\alpha_{i}|}{d},\dfrac{1}{d}\right\\}\quad\text{for}\quad
i=1,2,$

and let’s put

$b^{\prime}:=\dfrac{|b_{1}|}{d\log B_{2}}+\dfrac{|b_{2}|}{d\log B_{1}}.$

Let’s put

$\Gamma:=b_{1}\log\alpha_{1}+b_{2}\log\alpha_{2}.$

Note that $\Gamma\neq 0$ because $\alpha_{1}$ and $\alpha_{2}$ are
multiplicatively independent.

######  Théorème 2.5 (Laurent, Mignotte, Nesterenko).

With the previous notations, let $\alpha_{1}$, $\alpha_{2}$ be
multiplicatively independent positive numbers, then:

$\log|\Gamma|>-24.34d^{4}\left(\max\left\\{\log
b^{\prime}+0.14,\dfrac{21}{d},\dfrac{1}{2}\right\\}\right)^{2}\log B_{1}\log
B_{2}.$

Note that with $\Gamma:=b_{1}\log\alpha_{1}+b_{2}\log\alpha_{2}$, we have
$e^{\Gamma}-1=\Lambda,$ where
$\Lambda:=\alpha_{1}^{b_{1}}\cdots\alpha_{n}^{b_{n}}$ in case $n=2.$

##  3 Reduction method

During calculations, we get upper bounds on our variables which are too large,
so we have to reduce them. To do this, we use some results from the theory of
continued fractions. Concerning the treatment of homogeneous linear forms in
two integer variables, we use the well-known method of the classical result in
the theory of Diophantine approximation.

######  Lemme 3.1 (Legendre).

Let $\tau$ be an irrational number,
$\dfrac{p_{0}}{q_{0}},\dfrac{p_{1}}{q_{1}},\dfrac{p_{2}}{q_{2}},\ldots$ all
the convergents of the continued fraction of $\ tau$, and $M$ a positive
integer. Let $N$ be a positive integer such that $q_{N}>M.$ Then setting
$a(M):=\\{a_{i}:i=0,1,2,\ldots,N\\},$ the inequality

$\left|\tau-\dfrac{r}{s}\right|>\dfrac{1}{(a(M)+2)s^{2}},$

is valid for all pairs $(r,s)$ of positive integers with $0<s<M.$

For a non-homogeneous linear form with two integer variables, we use a slight
variation of a result due to Dujella and Pethő ([2], Lemma 5a). The proof is
almost identical to that of the corresponding result in [2]. For a real number
$X$, we write $\left\lVert X\right\rVert:=\min\\{\mid
X-n\mid:n\in\mathbb{Z}\\}$ for the distance from $X$ to the nearest integer.

######  Lemme 3.2 (Dujella, Pethő).

Let $M$ be a positive integer, $\dfrac{p}{q}$ a convergent of the continued
fraction of the irrational number $\tau$ such that $q>6M$, and $A,B,\mu$ be
numbers algebraic such as $A>0$ and $B>1.$ Also, $\varepsilon:=\left\lVert\mu
q\right\rVert-M\left\lVert\tau q\right\rVert$. If $\varepsilon>0$, then the
following inequality:

$0<|u\tau-v+\mu|<AB^{-w},$

does not admit an integer solution $u$, $v$ and $w$ with

$u\leq M\quad\text{et}\quad w\geq\dfrac{\log(Aq/\varepsilon)}{\log B}.$

On various occasions we need to find a lower bound for linear forms of
logarithms with bounded integer coefficients in three and four variables. In
this case, we use the Lenstra-Lenstra-Lovász basic lattice reduction algorithm
(LLL-algorithm) which we describe below. Let
$\tau_{1},\tau_{2},\ldots,\tau_{t}\in\mathbb{R}$ and the linear form

$x_{1}\tau_{1}+x_{2}\tau_{2}+\cdots+x_{t}\tau_{t}\quad\text{with}\quad|x_{i}|\leq
X_{i}.$ (16)

We set $X:=\max\\{X_{i}\\}$, $C>(tX)^{t}$ and consider the entire lattice
$\Omega$ generated by:

$b_{j}:=e_{j}+\lfloor c\tau_{j}\rceil\quad\text{for}\quad 1\leq j\leq
t-1\quad\text{and}\quad b_{t}:=\lfloor C\tau_{t}\rceil e_{t},$

where $C$ is a sufficiently large positive constant.

######  Lemme 3.3 (LLL-algorithme).

Let $X_{1},X_{2},\cdots,X_{t}$ be positive integers such that
$X:=\max\\{X_{i}\\}$ and $C>(tX)^{t}$ is a sufficiently large fixed positive
constant. With the above notations on the lattice $\Omega$, we consider a
reduced basis ${b_{i}}$ to $\Omega$ and its associated Gram-Schmidt
orthogonalization basis $\\{b_{i}^{*}\\}$. We fix

$\displaystyle c_{1}:=\max_{1\leq i\leq t}\dfrac{\left\lVert
b_{1}\right\rVert}{\left\lVert
b_{i}^{*}\right\rVert},\quad\theta:=\dfrac{\left\lVert
b_{1}\right\rVert}{c_{1}},\quad
Q:=\sum_{i=1}^{t-1}X_{i}^{2},\quad\text{et}\quad
R:=\dfrac{1}{2}\left(1+\sum_{i=1}^{t}X_{i}\right).$

If the integers $x_{i}$ are such that $|x_{i}|\leq X_{i}$, for $1\leq i\leq t$
and $\theta^{2}\geq Q+R^{2}$, then we have

$\left|\sum_{i=1}^{t}x_{i}\tau_{i}\right|\geq\dfrac{\sqrt{\theta^{2}-Q}-R}{C}.$

For proof and further details, we refer the reader to Cohen’s book.
(Proposition $2.3.20$ in [8], pp. $58-63$).

##  4 Main result

Suppose there are positive integers $n,m,n_{1},m_{1}$ such that
$(n,m)\neq(n_{1},m_{1}),$ and

$L_{n}-\mathcal{P}_{m}=L_{n_{1}}-\mathcal{P}_{m_{1}}.$

Due to symmetry, we can assume that $m\geq m_{1}$. If $m=m_{1}$, then
$L_{n}=L_{n_{1}}$, thus $(n,m)=(n_{1},m_{1})$, contradicting our hypothesis.
Thus, $m>m_{1}$. Seen that

$L_{n}-L_{n_{1}}=P_{m}-P_{m_{1}},$ (17)

and the right member is positive, we obtain that the left member is also
positive and therefore $n>n_{1}$. Thus, $n\geq 2$ and $n_{1}\geq 1$. Using
Binet’s formulas (11) and (7), the equation (17) implies that

$\displaystyle{\delta}^{n-3}\leq L_{n-2}\leq
L_{n}-L_{n_{1}}=\mathcal{P}_{m}-\mathcal{P}_{m_{1}}<\alpha^{m-1},$ (18a)
$\displaystyle{\delta}^{n+1}\geq
L_{n}>L_{n}-L_{n_{1}}=\mathcal{P}_{m}-\mathcal{P}_{m_{1}}\geq\mathcal{P}_{m-5}\geq\alpha^{m-7},$
(18b)

Hence

$(m-7)\left(\dfrac{\log\alpha}{\log{\delta}}\right)-1<n<(m-1)\left(\dfrac{\log\alpha}{\log{\delta}}\right)+3,$
(19)

where $\dfrac{\log\alpha}{\log{\delta}}=0.5843\ldots.$ If $n<$300, then $m\leq
190$. We ran a computer program for $2\leq n_{1}<n\leq 300$ and $1\leq
m_{1}<m<190$ and found only solutions from the list (LABEL:pllist). From now
on we assume that $n\geq 300$.
Note that the inequality (19) implies that $m<2n$. So, to solve the equation
(17), we need an upper bound on $n$.

###  4.1 Upper bound on n

Note that using the numerical inequalities(9) we have

$|\eta|^{n}+|\eta|^{n_{1}}+|b||\beta|^{m}+|c||\gamma|^{m}+|b||\beta|^{m_{1}}+|c||\gamma|^{m_{1}}<3.02.$
(20)

Using Binet’s formulas in the Diophantine equation (17), we get

$\begin{array}[]{rcl}\left|{\delta}^{n}-a\alpha^{m}\right|&=&\left|-\eta^{n}+{\delta}^{n_{1}}+\eta^{n_{1}}+(b\beta^{m}+c\gamma^{m})-(a\alpha^{m_{1}}+b\beta^{m_{1}}+c\gamma^{m_{1}})\right|\vspace{1mm}\\\
&\leq&{\delta}^{n_{1}}+a\alpha^{m_{1}}+|\eta|^{n}+|\eta|^{n_{1}}+|b||\beta|^{m}+|c||\gamma|^{m}+|b||\beta|^{m_{1}}+|c||\gamma|^{m_{1}}\vspace{1mm}\\\
&<&{\delta}^{n_{1}}+a\alpha^{m_{1}}+3.02\vspace{1mm}\\\
&<&4.76\max\\{{\delta}^{n_{1}},\alpha^{m_{1}}\\}.\end{array}$

By dividing by $a\alpha^{m}$ and using the relation (18a), we obtain

$\begin{array}[]{lll}\left|a^{-1}{\delta}^{n}\alpha^{-m}-1\right|&<&\max\left\\{\dfrac{4.76}{a\alpha^{m}}{\delta}^{n_{1}},\dfrac{4.76}{a}\alpha^{m_{1}-m}\right\\}\vspace{1mm}\\\
&<&\max\left\\{5.01\dfrac{{\delta}^{n_{1}}}{\alpha\cdot{\delta}^{n-3}},5.01\alpha^{m_{1}-m}\right\\}.\end{array}$

Therefore, we get

$\left|a^{-1}{\delta}^{n}\alpha^{-m}-1\right|<\max\\{{\delta}^{n_{1}-n+7},\alpha^{m_{1}-m+7}\\}.$
(21)

For the left member, we apply the theorem 2.3 with the data

$s=3,\quad\gamma_{1}=a,\quad\gamma_{2}=\delta,\quad\gamma_{3}=\alpha,\quad
b_{1}=-1,\quad b_{2}=n,\quad b_{3}=-m.$

Throughout our demonstrations, we work with
$\mathbb{K}:=\mathbb{Q}(\sqrt{5},\alpha)$ with $D=6$. Since
$\max\\{1,n,m\\}\leq 2n$ we take $B:=2n$. We have

$h(\gamma_{2})=\dfrac{\log{\delta}}{2}\quad\text{and}\quad
h(\gamma_{3})=\dfrac{\log\alpha}{3}.$

Moreover, the minimal polynomial of $\gamma_{1}$ is $23x^{3}-23x^{2}+6x-1$ and
has roots $a$, $b$, $c$. Since $|a|<1$ and $|b|=|c|<$1, then

$h(\gamma_{1})=\dfrac{1}{3}\log 23.$

So we can take

$A_{1}=2\log 23,\quad A_{2}=3\log{\delta},\quad A_{3}=2\log\alpha.$

Put

$\Lambda=a^{-1}{\delta}^{n}\alpha^{-m}-1.$

If $\Lambda=0$, then ${\delta}^{n}(\alpha^{-1})^{m}=a$, which is false, Since
${\delta}^{n}(\alpha^{-1})^{m}\in\mathcal{O}_{\mathbb{K}}$ while $a$ does not
belong to $\mathcal{O}_{\mathbb{K}}$, as can be seen immediately at from its
minimal polynomial. Thus, $\Lambda\neq 0$. Then, by theorem 2.3, the left side
of the equation (21) is bounded by

$\log\left|\Lambda\right|>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 6^{2}(1+\log
6)(1+\log 2n)(2\log 23)(3\log{\delta})(2\log\alpha).$

By comparing with (21), we get

$\min\\{(n-n_{1}-7)\log{\delta},(m-m_{1}-7)\log\alpha\\}<7.33\times
10^{13}(1+\log 2n),$

which gives

$\min\\{(n-n_{1})\log{\delta},(m-m_{1})\log\alpha\\}<7.33\times 10^{13}(1+\log
2n).$

Now two cases arise.
Case 1.
$\min\\{(n-n_{1})\log{\delta},(m-m_{1})\log\alpha\\}=(n-n_{1})\log{\delta}$.

In this case, we rewrite (17) as

$\left|\left({\delta}^{n-n_{1}}-1\right){\delta}^{n_{1}}-a\alpha^{m}\right|=\left|-a\alpha^{m_{1}}+\eta^{n}-\eta^{n_{1}}+(b\beta^{m}+c\gamma^{m})-(b\beta^{m_{1}}+c\gamma^{m_{1}})\right|$

using (20) and dividing by $\alpha^{m}$, we get

$\left|\left(\dfrac{{\delta}^{n-n_{1}}-1}{a}\right){\delta}^{n_{1}}\alpha^{-m}-1\right|<5.21\alpha^{m_{1}-m}.$
(22)

We put

$\Lambda_{1}=\left(\dfrac{{\delta}^{n-n_{1}}-1}{a}\right){\delta}^{n_{1}}\alpha^{-m}-1.$

We see that, $\Lambda_{1}\neq 0$, for if $\Lambda_{1}=0$, then
${\delta}^{n}-{\delta}^{n_{1}}=a\alpha^{m}$. This is impossible because
$a\alpha^{m}\in\mathbb{Q}(\alpha)$ and
${\delta}^{n}-{\delta}^{n_{1}}\in\mathbb{Q}(\sqrt{5})\ backslash\mathbb{Q}$.
Indeed, if ${\delta}^{n}-{\delta}^{n_{1}}\in\mathbb{Q}$, then when we take
$\sigma\neq id$ to be the unique non-trivial $\mathbb{Q}$\- automorphism on
$\mathbb{Q}(\sqrt{5})$. Then we get

${\delta}^{n}-{\delta}^{n_{1}}=\sigma({\delta}^{n}-{\delta}^{n_{1}})=\eta^{n}-\eta^{n_{1}}.$

However, the absolute value of the left member is at least
${\delta}^{n}-{\delta}^{n_{1}}\geq{\delta}^{n-2}\geq{\delta}^{298}>2$, while
the absolute value on the right is at most
$\left|\eta^{n_{1}}-\eta^{n}\right|\leq|{\eta}|^{n_{1}}+|{\eta}|^{n}<2.$ By
this obvious contradiction, we conclude that $\Lambda_{1}\neq 0$.

We apply the lemma 2.3 by taking $s=3$, and

$\gamma_{1}=\dfrac{{\delta}^{n-n_{1}}-1}{a},\quad\gamma_{2}=\delta,\quad\gamma_{3}=\alpha,\quad
b_{1}=1,\quad b_{2}=n_{1},\quad b_{3}=-m.$

The minimal polynomial of ${\delta}^{n-n_{1}}+1$ divides

$x^{2}+(2-L_{n-n_{1}})x+((-1)^{n-n_{1}}+1-L_{n-n_{1}}),$

where $\\{L_{k}\\}_{k\geq 0}$ is the Lucas sequence defined by $L_{0}=2$,
$L_{1}=1$, $L_{k+2}=L_{k+1}+L_{k}$ for all $k\geq 0$, for which the Binet
formula of its general term is

$L_{k}={\delta}^{k}+\eta^{k}\quad\text{for all }k\geq 0.$

On the other hand, the minimal polynomial of $a$ is $23x^{3}-23x^{2}+6x-1$ and
has roots $a$, $b$, $c$. Since $|b|=|c|<1$ and $a<1$, then $h(a)=\dfrac{\log
23}{3}$.

Thus, we get

$\begin{array}[]{rcl}h(\gamma_{1})&\leq&h\left({\delta}^{n-n_{1}}+1\right)+h(a)\vspace{2mm}\\\
&\leq&h\left({\delta}^{n-n_{1}}\right)+h(1)+\log 2+\dfrac{\log
23}{3}\vspace{2mm}\\\ &\leq&(n-n_{1})h\left({\delta}\right)+h(1)+\log
2+\dfrac{\log 23}{3}\vspace{2mm}\\\
&<&\dfrac{1}{2}(n-n_{1})\log\delta+1.74\vspace{2mm}\\\ &<&3.66\times
10^{13}(1+\log 2n).\end{array}$ (23)

Thus, we can take $A_{1}:=2.2\times 10^{14}(1+\log 2n)$. Also, as before, we
can take $A_{2}:=3\log{\delta}$ and $A_{3}:=2\log\alpha$. Finally, since
$\max\\{1,n_{1},m\\}\leq 2n$, we can take $B:=2n$. We then obtain

$\log\left|\Lambda_{1}\right|>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 6^{2}(1+\log
6)(1+\log 2n)\times(2.2\times 10^{14}(1+\log
2n))(3\log{\delta})(2\log\alpha).$

Thereby,

$\log\left|\Lambda_{1}\right|>-2.57\cdot 10^{27}(1+\log 2n)^{2}.$

Comparing this with (22), we get that

$(m-m_{1})\log\alpha<2.57\cdot 10^{28}(1+\log 2n)^{2}.$

Case 2.
$\min\\{(n-n_{1})\log{\delta},(m-m_{1})\log\alpha\\}=(m-m_{1})\log\alpha$.

In this case, we rewrite (17) as

$\left|{\delta}^{n}-a\alpha^{m}+a\alpha^{m_{1}}\right|=\left|\eta^{n}+{\delta}^{n_{1}}-\eta^{n_{1}}+(b\beta^{m}+c\gamma^{m})-(b\beta^{m_{1}}+c\gamma^{m_{1}})\right|$

then

$\left|\dfrac{{\delta}^{n}\alpha^{-m_{1}}}{a(\alpha^{m-m_{1}}-1)}-1\right|<\dfrac{4.03}{a(1-\alpha^{m_{1}-m})\alpha}\dfrac{{\delta}^{n_{1}}}{\alpha^{m-1}}<18{\delta}^{n_{1}-n+3}.$
(24)

Let

$\Lambda_{2}=(a(\alpha^{m-m_{1}}-1))^{-1}{\delta}^{n}\alpha^{-m_{1}}-1.$

We see that, $\Lambda_{2}\neq 0$, for if $\Lambda_{2}=0$ implies
${\delta}^{2n}=\alpha^{2m_{1}}a^{2}(\alpha^{m-m_{1}}-1)^{2}.$, however,
${\delta}^{2n}\in\mathbb{Q}(\sqrt{5})\backslash\mathbb{Q}$, while
$\alpha^{2m_{1}}a^{2}(\alpha^{m-m_{1}}-1)^{2}\in\mathbb{Q}(\alpha)$, which is
not possible. We apply again the lemma 2.3. In this application, we take
$s=3$, and

$\gamma_{1}=a(\alpha^{m-m_{1}}-1),\quad\gamma_{2}=\delta,\quad\gamma_{3}=\alpha,\quad
b_{1}=-1,\quad b_{2}=n,\quad b_{3}=-m_{1}.$

We have

$\begin{array}[]{lll}h(\alpha^{m-m_{1}}-1)&\leq&h(\alpha^{m-m_{1}})+h(-1)+\log
2=(m-m_{1})h(\alpha)+\log 2\vspace{1mm}\\\
&=&\dfrac{(m-m_{1})\log\alpha}{3}+\log 2<2.43\times 10^{13}(1+\log 2n)+\log
2.\end{array}$

Thus, on

$\begin{array}[]{lll}h(\gamma_{1})&<&2.44\times 10^{13}(1+\log 2n)+\log
2+\dfrac{\log 23}{3}+\log\sqrt{5}\vspace{1mm}\\\ &<&2.44\times 10^{13}(1+\log
2n).\end{array}$

So we can take $A_{1}:=1.47\times 10^{14}(1+\log 2n)$. Also, as before, we can
take $A_{2}:=3\log{\delta}$ and $A_{3}:=2\log\alpha$. Finally, since
$\max\\{1,n,m_{1}+1\\}\leq 2n$, we can take $B:=2n$. We then get this

$\log\left|\Lambda_{2}\right|>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 6^{2}(1+\log
6)(1+\log 2n)\times(1.47\times 10^{14}(1+\log
2n))(3\log{\delta})(2\log\alpha).$

Hence,

$\log\left|\Lambda_{1}\right|>-1.71\cdot 10^{27}(1+\log 2n)^{2}.$

Comparing this with (24), we get that

$(n-n_{1})\log{\delta}<1.71\cdot 10^{27}(1+\log 2n)^{2}.$

Thus, in both cases 1 and 2, we have

$\displaystyle\min\\{(n-n_{1})\log{\delta},(m-m_{1})\log\alpha\\}<$
$\displaystyle 7.33\cdot 10^{13}(1+\log 2n)$ (25a)
$\displaystyle\max\\{(n-n_{1})\log{\delta},(m-m_{1})\log\alpha\\}<$
$\displaystyle 2.57\cdot 10^{27}(1+\log 2n)^{2}.$ (25b)

We finally rewrite the equation (17) as

$\left|{\delta}^{n}-{\delta}^{n_{1}}-a\alpha^{m}+a\alpha^{m_{1}}\right|=\left|\delta^{n}-\delta^{n_{1}}+(b\beta^{m}+c\gamma^{m})-(b\beta^{m_{1}}+c\gamma^{m_{1}})\right|<3$

Dividing both sides by $a\alpha^{m_{1}}(\alpha^{m-m_{1}}-1)$, we get

$\left|\left(\dfrac{{\delta}^{n-n_{1}}-1}{a(\alpha^{m-m_{1}}-1)}\right){\delta}^{n_{1}}\alpha^{-m_{1}}-1\right|<\dfrac{3}{a(1-\alpha^{m_{1}-m})\alpha}\dfrac{1}{\alpha^{m-1}}<13.5{\delta}^{3-n}.$
(26)

To find a lower bound on the left side, we again use the lemma 2.3 with $s=3$,
and

$\gamma_{1}=\dfrac{{\delta}^{n-n_{1}}-1}{a(\alpha^{m-m_{1}}-1)},\quad\gamma_{2}=\delta,\quad\gamma_{3}=\alpha,\quad
b_{1}=1,\quad b_{2}=n_{1},\quad b_{3}=-m_{1}.$

Using $h(x/y)=h(x)+h(y)$ for two nonzero algebraic numbers $x$ and $y$, we
have

$\begin{array}[]{rcl}h(\gamma_{1})&\leq&h\left(\dfrac{{\delta}^{n-n_{1}}-1}{a}\right)+h(\alpha^{m-m_{1}}-1)\vspace{1mm}\\\
&<&\dfrac{1}{2}(n-n_{1}+4)\log{\delta}+\dfrac{\log
23}{3}+\dfrac{(m-m_{1})\log\alpha}{3}+\log 2\vspace{1mm}\\\ &<&2.14\cdot
10^{27}(1+\log 2n)^{2},\end{array}$

where in the chain of inequalities above, we used the argument of(23) as well
as the (25b) bound. So we can take $A_{1}:=1.54\cdot 10^{28}(1+\log 2n)^{2}$
and certainly $A_{2}:=3\log{\delta}$ and $A_{3}:=2\log\ alpha$. Using
arguments similar to those in the proof that $\Lambda_{1}\neq 0$ we show that
if we set

$\Lambda_{3}=\left(\dfrac{{\delta}^{n-n_{1}}-1}{a(\alpha^{m-m_{1}}-1)}\right){\delta}^{n_{1}}\alpha^{-m_{1}}-1,$

then $\Lambda_{3}\neq 0$. The Lemma 2.3 gives

$\log\left|\Lambda_{3}\right|>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 6^{2}(1+\log
6)(1+\log 2n)\times(1.54\cdot 10^{28}(1+\log
2n)^{2})(3\log{\color[rgb]{0,0,1}{\delta}})(2\log\alpha),$

which with (26) gives

$(n-3)<1.8\cdot 10^{41}(1+\log 2n)^{3},$

leading to $n<2.45\cdot 10^{47}$.

###  4.2 Reduction of the upper bound by n

We must now reduce the bound above for $n$ and to do this we use the lemma 3.2
several times and each time $M:=2.45\cdot 10^{47}$. To begin, let’s go back to
(21) and set

$\Gamma:=n\log{\delta}-m\log\alpha-\log a.$

For technical reasons, we assume that $\min\\{n-n_{1},m-m_{1}\\}\geq 20$.
Let’s go back to the inequalities for $\Lambda,\leavevmode\nobreak\
\Lambda_{1},\leavevmode\nobreak\ \Lambda_{2}$.

Since we assume that $\min\\{n-n_{1},m-m_{1}\\}\geq 20$, we get
$|e^{\Gamma}-1|=|\Lambda|<\dfrac{1}{4}$. However, $|\Lambda|<\dfrac{1}{2}$ and
since the inequality $|x|<2|e^{x}-1|$ holds for all
$x\in\left(-\frac{1}{2},\frac{1}{2}\right)$, we get

$\left|\Gamma\right|<2\max\\{{\delta}^{n_{1}-n+7},\alpha^{m_{1}-m+7}\\}\leq\max\\{{\delta}^{n_{1}-n+9},\delta^{m_{1}-m+10}\\}.$

Assume $\Gamma>0$. We then have the inequality

$\begin{array}[]{rcl}0<n\left(\dfrac{\log{\delta}}{\log\alpha}\right)-m+\dfrac{\log(1/a)}{\log\alpha}&<&\max\left\\{\dfrac{{\delta}^{9}}{\log\alpha}{\delta}^{-(n-n_{1})},\dfrac{\alpha^{10}}{\log\alpha}\alpha^{-(m-m_{1})}\right\\}\vspace{2mm}\\\
&<&\max\\{270\cdot{\delta}^{-(n-n_{1})},60\cdot\alpha^{-(m-m_{1})}\\}.\end{array}$

We apply the lemma 3.2 with

$\tau=\dfrac{\log{\delta}}{\log\alpha},\quad\mu=\dfrac{\log(1/a)}{\log\alpha},\quad(A,B)=(270,\delta)\;\text{
or }\;(60,\alpha).$

Let $\tau=[a_{0},a_{1},\ldots]=[1;1,2,2,6,2,1,2,1,2,1,1,11,\ldots]$ is the
continued fraction of $\tau$. We consider the convergent $98$-th

$\dfrac{p}{q}=\dfrac{p_{98}}{q_{98}}=\dfrac{(78093067704223831799032754534503501859635391435517}{45634243076387457097046528084208490147594968308975}$

which satisfies $q=q_{98}>6M$. Moreover, this gives $\varepsilon>0.37$, and
therefore either

$n-n_{1}\leq\dfrac{\log(270q/\varepsilon)}{\log{\delta}}<250,\;\text{ or
}\;m-m_{1}\leq\dfrac{\log(60q/\varepsilon)}{\log\alpha}<420.$

In the case of $\Gamma<0$, we consider the following inequality:

$\begin{array}[]{rcl}m\left(\dfrac{\log\alpha}{\log{\delta}}\right)-n+\dfrac{\log
a}{\log{\delta}}&<&\max\left\\{\dfrac{{\delta}^{9}}{\log{\delta}}\alpha^{-(n-n_{1})},\dfrac{\alpha^{10}}{\log{\delta}}\alpha^{-(m-m_{1})}\right\\}\vspace{2mm}\\\
&<&\max\\{160\cdot{\delta}^{-(n-n_{1})},37\cdot\alpha^{-(m-m_{1})}\\},\end{array}$

instead and apply the lemma 3.2 with

$\tau=\dfrac{\log\alpha}{\log{\delta}},\quad\mu=\dfrac{\log
a}{\log{\delta}},\quad(A,B)=(160,\delta)\;\text{ or }\;(35,\alpha).$

Let $\tau=[a_{0},a_{1},\ldots]=[0,1,1,2,2,6,2,1,2,1,2,1,1,11,\ldots]$ be the
fraction sequence of $\tau$ (note that the current $\tau$ is just the inverse
of the previous $\tau$). Again, we consider the convergent $98$-th that
satisfies $q=q_{98}>6M$. This again gives $\varepsilon>0.0867$, and so either

$n-n_{1}\leq\dfrac{\log(160q/\varepsilon)}{\log{\delta}}<246<250,\;\text{ or
}\;m-m_{1}\leq\dfrac{\log(35q/\varepsilon)}{\log\alpha}<413<420.$

In conclusion, we have either $n-n_{1}\leq 250$ or $m-m_{1}\leq 420$ whenever
$\Gamma\neq 0$.

Now, we must distinguish the cases $n-n_{1}\leq 250$ and $m-m_{1}\leq 420$.
First suppose that $n-n_{1}\leq 250$. In this case, we consider the inequality
(22) and assume that $m-m_{1}\geq 20$. We ask

$\Gamma_{1}=n_{1}\log{\delta}-m\log\alpha+\log\left(\dfrac{{\delta}^{n-n_{1}}-1}{a}\right).$

Then the inequality (22) implies that

$\left|\Gamma_{1}\right|<10.4\alpha^{m_{1}-m}.$

If we further assume that $\Gamma_{1}>0$, then we get

$0<n_{1}\left(\dfrac{\log{\delta}}{\log\alpha}\right)-m+\dfrac{\log(({\delta}^{n-n_{1}}-1)/a)}{\log\alpha}<\dfrac{10.4}{(\log\alpha)}\alpha^{-(m-m_{1})}<38\alpha^{-(m-m_{1})}.$

We apply again the lemma 3.2 with the same $\tau$ as in the case where
$\Gamma>0$. We use the $100$-th ${p}/{q}={p_{98}}/{q_{98}}$ convergent to
$\tau$ as before. But in this case we choose $(A,B):=(30,\alpha)$ and use

$\mu_{k}=\dfrac{\log(({\delta}^{k}-1)/a)}{\log\alpha},$

instead of $\mu$ for each possible value of $k:=n-n_{1}\in[1,2,\ldots 250].$
For the remaining values of $k$, we get $\varepsilon>0.00292$. Thus, according
to the lemma 3.2, we obtain

$m-m_{1}<\dfrac{\log(38q/0.00292)}{\log\alpha}<441.$

Thus, $n-n_{1}\leq 250$ implies $m-m_{1}\leq 441$. In the case where
$\Gamma_{1}<0$ we follow the ideas of the case where $\Gamma_{1}>0$. We use
the same $\tau$ as in the case where $\Gamma<0$ but instead of $\mu$ we take

$\mu_{k}=\dfrac{\log(a/({\delta}^{k}-1))}{\log{\delta}},$

for each possible value of $n-n_{1}=k=1,2,\ldots,250$. By using the lemma 3.2
with this parameter we also obtain in this case that $n-n_{1}\leq 250$ implies
$m-m_{1}\leq 435$.

In conclusion for $n-n_{1}\leq 250$ we have $m-m_{1}\leq 441$. Now let’s go to
the case where $m-m_{1}\leq 420$ and consider the inequality (24). we put

$\Gamma_{2}=n\log{\delta}-m_{1}\log\alpha+\log(1/(a(\alpha^{m-m_{1}}-1))),$

and we assume that $n-n_{1}\geq 20$. We then have

$\left|\Gamma_{2}\right|<\dfrac{36.1{\delta}^{4}}{\delta^{n-n_{1}}}.$

Assuming $\Gamma_{2}>0$, we get

$0<n\left(\dfrac{\log{\delta}}{\log\alpha}\right)-m_{1}+\dfrac{\log((1/(a(\alpha^{m-m_{1}}-1)))}{\log\alpha}<540\cdot{\delta}^{-(n-n_{1})}.$

We apply the lemma 3.2 again with the same $\tau$, $q$, $M$,
$(A,B):=(540,\delta)$ and

$\mu_{k}=\dfrac{\log(1/(a(\alpha^{k}-1))}{\log\alpha}\quad\text{for
}k=1,2,\ldots 420.$

We obtain $\varepsilon>0.000354$, thus

$n-n_{1}<\dfrac{\log(540q/0.000354)}{\log{\delta}}<256.$

A similar conclusion is reached when $\Gamma_{2}<0$, indeed We get
$\varepsilon>0.000508$, so

$n-n_{1}<\dfrac{\log(320q/0.000508)}{\log{\delta}}<256.$

In conclusion, for $m-m_{1}\leq 420$ we have $n-n_{1}\leq 256$. So
$m-m_{1}\leq 441$ and $n-n_{1}\leq 256$. Finally, we go to (26). we put

$\Gamma_{3}=n_{1}\log{\delta}-m_{1}\log\alpha+\log\left(\dfrac{{\delta}^{n-n_{1}}-1}{a(\alpha^{m-m_{1}}-1)}\right).$

Since $n\geq 200$, the inequality (26) implies that

$\left|\Gamma_{3}\right|<\dfrac{16.5}{{\delta}^{n-4}}=\dfrac{27{\delta}^{4}}{\delta^{n}}.$

Suppose $\Gamma_{3}>0$. Then

$0<n_{1}\left(\dfrac{\log{\delta}}{\log\alpha}\right)-m_{1}+\dfrac{\log(({\delta}^{k}-1)/(a(\alpha^{l}-1)))}{\log\alpha}<240\cdot{\delta}^{-n},$

where $(k,l):=(n-n_{1},m-m_{1})$. We apply the lemma 3.2 again with the same
$\tau=\dfrac{\log{\delta}}{\log\alpha}$, $q_{98}$, $(A,B):=(240,\delta)$ and

$\mu_{k,l}=\dfrac{\log(({\delta}^{k}-1)/(a(\alpha^{l}-1)))}{\log\alpha}\quad\text{for
}1\leq k\leq 256,\;1\leq l\leq 441.$

We consider the $98$-th $\dfrac{p_{98}}{q_{98}}$ convergent. For all pairs
$(k,l)$ we get that $\varepsilon>1.43\times 10^{-6}$. Thus, the lemma 3.2
shows that

$n<\dfrac{\log(240\times q_{98}\times 10^{6}/1.43)}{\log{\delta}}<271.$

The theorem 1.1 is therefore proven.

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# Understanding and visualizing the statistical analysis of SN1987A neutrino
data

Marcos V. dos Santosa and Pedro Cunha de Holandaa
aInstituto de Física Gleb Wataghin - Unicamp

###### Abstract

The SN1987A detection through neutrinos was an event of great importance in
neutrino physics, being the first detection of neutrinos created outside our
solar system, and then inaugurating the era of experimental neutrino
astronomy. The data have been largely studied in many different analysis, and
has presented several challenges in different aspects, since both supernova
explosion dynamics and neutrino flavour conversion in such extreme environment
still have many unknowns. In addition, the low statistics also invoke the need
of unbinned statistical methods to compare any model proposal with data. In
this paper we focus on a discussion about the most used statistical analysis
interpretation, presenting a pedagogical way to understand and visualize this
comparison.

Keywords: _supernova_ ; _neutrino_.

##  1 Introduction

Particle physics provides a fertile ground to a vast number of methods to
statistically compare theory and data, giving a quantitative filling in order
to guide the prospects of a theory, or even reveal important accomplishments
or tensions in experimental efforts.

In general, the theory can visually (and intuitively) be compared to data for
most of the methods, but one interesting phenomena, the neutrino burst
detected from the supernova SN1987A, that is particularly affected by low
statistics, evidence difficulties to such parallel.

In this paper we propose a visual understanding to handle this difficulty
through an animation of a parameterized model and SN1987A antineutrino data.

##  2 Statistical Analysis

The quantitative comparison between theoretical predictions and experimental
results is a major part of any scientific endeavour. As in many other areas,
in particle physics an important quantity around such comparison is made is
the detection rate of a specific event. As an example, theoretical models of
solar neutrino production provide us with a steady theoretical neutrino flux.
Solar neutrino experiments provides us with a detected event rate. The
comparison between these two quantities can be done by translating the
theoretical flux in an expected event rate, or inversely, translating the
detected event rate in a compatible expected flux.

Besides, a lot of information can be extracted from the dependence of neutrino
flux with its energy or time of detection. The most straightforward way to
include this information on statistical analysis would be to split the total
data into bins of specific energy or time intervals. Maybe the most spread
statistical tool to implement these kind of analysis would be to calculate the
following $\chi^{2}$:

$\chi^{2}=\sum_{i,j}(R^{th}_{i}-R^{ex}_{i})(\sigma^{-2})_{ij}(R^{th}_{j}-R^{ex}_{j})$
(1)

where the indices $i$ and $j$ track the binning of the data, $R^{th}_{k}$ is
the theoretical prediction and $R^{ex}_{k}$ is the experimental data. This
analysis has one great advantage: it allows to visually grasp how good is the
accordance between theory and experiment in a figure where experimental data
points, uncertainties and theoretical predictions can be plotted together. If
theoretical predictions are contained inside the region around the
experimental data points delimited by the uncertainties, than we expect a good
fit.

Several examples in neutrino physics can illustrate such procedure. Again
taking solar neutrinos as an example, the data presented by the neutrino
detector Super-Kamiokande is divided both in energy and time bins. In Figure 1
the data of 1496 days of running experiment are presented, and the binning on
energy and time can be seen, although the binning in time is indirect, through
a binning in the position of the Sun when the data was taken. The continuous
line represent the prediction for the best fit point of the statistical
analysis when flavour conversion is considered in two scenarios, the LMA and
the LOW solutions.

Figure 1: Data from the 1496 days of Super-Kamiokande. Each day/night plot
gives a binning in energy, LOW (light gray) and LMA (black) proposed solutions
to be visually compared to data points. Taken from [1].

As discussed, it is possible to visually get a feeling about how good the
accordance between experimental data and theoretical predictions by how the
theoretical curves cross the region around the experimental data within the
error bars. For this particular example, two solutions to the solar neutrino
problem are presented, with large mixing angle and large $\Delta m^{2}$ (LMA)
and lower $\Delta m^{2}$ (LOW). We can expect by visually inspecting the
figure that both solutions would fit the data quite well, fact that is
confirmed by a more careful statistical analysis.

The problem of this visualization is when there is no efficient way to collect
the data to form bins, for instance, due to the low event rate. In particular,
when the event rate is very low it is necessary to take the experimental data
event by event.

In this context, what we propose here is to recover a way to visually access
how good the accordance between experimental data and theoretical prediction
in a particular scenario when the statistical analysis is done event by event:
the neutrino data from Supernova 1987A.

##  3 Supernova 1987A

The Core Collapse Supernova is a remarkable end of life of a star and one of
the most peculiar astrophysical phenomena. Despite being a prominent optical
phenomena, its most outstanding property is the powerful release of $\sim
99\%$ of gravitational binding energy, generally in the order of $\sim
10^{53}$ erg, from a $m\gtrsim 10M_{\odot}$ progenitor star in (anti)neutrinos
of all flavors in MeV scale.

Nevertheless the high neutrino luminosity, a limitation in the neutrino
observation on Earth is related to the high distance $D$ from the source, with
decreasing flux proportional to $D^{-2}$, restricting the possible region for
neutrino burst detection to a galactic or nearby the Milky Way, that possesses
a low supernova rate of $\sim 1$ per century [2].

Even though, in 1987, three detectors, Kamiokande II [3], IMB [4, 5] and
Baksan [6], were capable to observe a neutrino signal associated to a SN in
the Large Magellanic Cloud ($\sim 50$ kpc). These data are presented in Figure
2. In contrast with Super-Kamiokande data in Figure 1, these are individual
events, and there is no obvious way to overlap a theoretical curve to them.
Since the theoretical models provide a flux density and any kind of binning to
convert this density into a event probability would be quite arbitrary, the
approach through an unbinned maximum likelihood estimation is a robust
alternative to confront the theoretical hypothesis with these individual
events. In next section we describe such procedure.

Figure 2: Positron energy and relative time from the Kamiokande II, IMB and
Baksan detectors, with a total of 29 events.

##  4 Modelling SN1987A event-by-event likelihood

Frequently the likelihood treatment in particle physics involves the usage of
Poisson distribution $P(\mu,n)$, that fits well to phenomena that has small
probability to occur, but a large number of tries. Given a measured variable
set $\vec{x}$, the Poisson likelihood is given by:

$\mathcal{L}=\prod_{i=1}^{N_{bins}}\frac{{\mu(x_{i})}^{n_{i}}}{n_{i}!}e^{\mu(x_{i})}$
(2)

where $n_{i}$ can be a particular number of events that occurs in a
$x_{i}+\delta x_{i}$ interval of our variable, in a number $N$ of intervals,
or bins, and $\mu(x_{i})$ is the expected value in the same interval. It is
convenient to write $\mu(x_{i})$ as a distribution function on the variable
$x_{i}$, or $\mu(x_{i})=R(x_{i})\delta x_{i}$, with a given events rate
$R(x_{i})=\frac{dN}{dx_{i}}$ in an equally spaced bin of variable $\delta
x_{i}$ and number of counts $n_{i}$. Including it in equation (2):

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{N_{bins}}\frac{[R(x_{i})\delta
x]^{n_{i}}}{n_{i}!}e^{-R(x_{i})\delta x}$ (3) $\displaystyle=$ $\displaystyle
e^{-\sum_{j=1}^{N_{bins}}R(x_{j})\delta
x}\prod_{i=1}^{N_{bins}}\frac{[R(x_{i})\delta x]^{n_{i}}}{n_{i}!}\;.$

However, binning the data to use a single expected value of a set of points
requires to assume a given statistical distribution of such a bin, that
generally is assumed to be Gaussian for higher number of entries. The low
statistics scenario does not allow this assumption, then it is possible to
model the likelihood (3) to account for each event apart. This can be made by
taking the bin to an infinitesimal width $\delta x\rightarrow dx$ and number
of counts $n_{i}\rightarrow 1$, so we consider only infinitesimal bins with
one event and drop the others, then (3) becomes

$\mathcal{L}\propto e^{-\int R(x)dx}\prod_{i=1}^{N_{obs}}R(x_{i})$ (4)

that also has the change from total number of bins $N_{bins}$ to total number
of observed events $N_{obs}$. The idea behind maximum likelihood is to
maximize the quantity in (4), or given the correspondence
$\mathcal{L}=e^{-\chi^{2}/2}$, minimize the
$\chi^{2}(\vec{x})=-2\log\mathcal{L}(\vec{x})$ to respect to a free set of
parameters $\vec{x}$. If we have a single event at $x=\bar{x}$, this
expressions reduces to $e^{-\int R(x)dx}R(\bar{x})$. For different models with
a normalized expected event rate $\int R(x)dx$, the likelihood is maximized
for the model with the highest value of $R(x_{i})$. And letting the
normalization runs freely, it is maximized for $\int R(x)dx=1$. It is
straightforward to note that if we consider more than one single event this
maximum occurs on the total number of events.

In a supernova detection, such as SN1987A, the variables $x$ are the neutrino
energy, the detection time and events scattering angle, i.e.
$R=R(t,E,\cos\theta)$. Then eq. (4) becomes:

$\mathcal{L}=e^{-\int
R(t,E,\cos\theta)\,dt\,dE\,d\cos\theta}\prod_{i=1}^{N_{obs}}R(t_{i},E_{i},\cos\theta_{i})\,dt\,dE\,d\cos\theta$
(5)

where $R$ is a triply differential equation,
$R=\frac{d^{3}N}{dt\,dE\,d\cos\theta}$ and $N$ is the expected number of
events at the detector. For simplicity we did not include the scattering angle
dependence on the animations presented in the following, although they were
used in the likelihood calculation. A complete analysis, including other
details such as background, energy resolution and efficiency can be seen in
[7], [8] and [9].

Figure 3: Theoretical events rate cumulative integrated over time (Eq. 6)
(blue) and normally distributed data as proposed in (7) (red) changing along
relative detection time since the first measured neutrino from SN1987A. The
time scale runs logarithmically in the first second and linearly afterwards to
better show the data structure for early time events. See the animation at:
https://github.com/santosmv/Animations-visualizing-SN1987A-data-
analysis/blob/main/events_rate.png.

##  5 Single event distribution

The main ingredient to construct the likelihood is the theoretical triply
differential expected rate. However, since there is no way to convert the
theoretical predictions into some quantity to be compared with individual
events, we can instead modify the events to match the theoretical probability
distribution. For instance, all SN1987A events are published with an
uncertainty in energy, so the true information we can take from each event is
a probability distribution around some most probable result. Assuming such
distribution to be Gaussian, a specific event with energy $\bar{E}_{\nu}$ and
uncertainty $\sigma_{E}$ on time $\bar{t}$ with uncertainty $\sigma_{t}$ is
related to the following probability distribution:

$\frac{d^{2}P(E_{\nu},t)}{dE_{\nu}\,dt}=\frac{1}{\sigma_{E}\sqrt{2\pi}}\exp{\left(-\frac{1}{2}\frac{(E_{\nu}-\bar{E}_{\nu})^{2}}{\sigma_{E}^{2}}\right)}\times\frac{1}{\sigma_{t}\sqrt{2\pi}}\exp{\left(-\frac{1}{2}\frac{(t-\bar{t})^{2}}{\sigma_{t}^{2}}\right)}$

This can be compared with the theoretical probability of inducing an event on
the detector:

$\frac{d^{2}N}{dE_{\nu}\,dt}=A\,\frac{d^{2}\phi(E_{\nu},t)}{dE_{\nu}\,dt}\,\sigma(E_{\nu})$
(6)

where $A$ is a proportionality factor that take into account the detector size
and efficiency.

To proper visualize the data points being collected, we can create an
animation with the detected event probability integrated on time. Since the
uncertainty on time is very small, the distribution converges to a
$\delta$-function, and such animation would advance in steps while the data
gets collected:

$\sum_{i}\frac{dP_{i}(E_{\nu},t)}{dE_{\nu}}=\int_{t_{0}}^{t}dt\,\sum_{i}\frac{d^{2}P_{i}(E_{\nu},t)}{dE_{\nu}\,dt}=\sum_{i}\frac{1}{\sigma_{E_{i}}\sqrt{2\pi}}\exp{\left(-\frac{1}{2}\frac{(E_{\nu}-\bar{E}_{\nu
i})^{2}}{\sigma_{E_{i}}^{2}}\right)}\theta(t-t_{i})$ (7)

Such animation is presented in Figure 3 (red curve). Since what is presented
is the cumulative result after integrating on time, the final moment of this
animation, when integrated also on energy, provides all the 29 events detected
by the three experiments. The comparison with theoretical predictions can be
made visually if we produce a similar animation for the expected number of
events, integrating Eq. (6) on time, also presented in Figure 3 (dashed
curve).

The parameterization of the antineutrino111Once the detectors in 1987 were
capable to measure a single channel, the inverse beta decay
($\bar{\nu}_{e}+p\rightarrow e^{+}+n$), only electron antineutrinos could be
detected. flux $\phi(E_{\nu},t)$ in Eq. (6) consists in a two-component
emission (accretion + cooling) and nine free parameters, that come from the
proposed flux $\phi=\phi(t,E,\cos\theta,\vec{y})$, with
$\vec{y}=(T_{c},R_{c},\tau_{c},T_{a},M_{a},\tau_{a})$, where $T_{c}$ ($T_{a}$)
is the initial antineutrino (positron) temperature from the cooling
(accretion) phase, $R_{c}$ is the radius of the neutrinosphere, $\tau_{c}$
($\tau_{a}$) is the characteristic time from the cooling (accretion) phase and
$M_{a}$ is the initial accreting mass [8]. The remaining three free parameters
are a time offset $t_{\textrm{off}}$ to be adjusted independently for each
detector.

These parameters are estimated from an event-by-event maximum likelihood, and
the best fit values of our analysis, used in Figure 3, are:

$\displaystyle
T_{c}=4.8\,\mbox{MeV},~{}R_{c}=6.7\,\mbox{km},~{}\tau_{c}=4.8\,\mbox{s},$ (8)
$\displaystyle
T_{a}=1.7\,\mbox{MeV},~{}M_{a}=0.4\,M_{\odot},~{}\tau_{a}=0.62\,\mbox{s}$ (9)

As described before, the maximization on the likelihood depends on two terms.
The term in the exponential factor factor is related to the number of events,
and drives the theoretical parameters to those who provides the right expected
number of events, i.e., the area under the curves in the end of the animation
in Figure 3. It is quite easy to grasp if our theoretical model fits well the
data by this aspect.

The second term access how close the theoretical curve is to the experimental
one at the data central points, both in energy and in time. Since the
uncertainty in time is negligible, we can visually compare the curves at the
moments a new data is collected, providing us with a visual tool to this
second ingredient of the statistical analysis. By performing these two
analysis on Figure 3, we can expect that, although not perfect, the
theoretical prediction would provide a reasonably good fit to the data.

Figure 4: To visualize the effect of spectral distortion impact on events
rate, we used two sets of parameters for $T_{a}$ and $M_{a}$ (in green and in
cyan) that are excluded at $90\%$ C.L. according to our analysis. The green
curve produces a distortion for low energy events, while the cyan produces a
distortion that favour high-energy events. Also on light grey we present the
best-fit point of our analyses, shown in Fig. 2. See the animation at:
https://github.com/santosmv/Animations-visualizing-SN1987A-data-
analysis/blob/main/events_rate_worse_fit.png

It is useful now to analyse a set of theoretical parameters that do not fit
well the data. This is done in Fig 4, where we chosed two set of parameters
that are excluded at 90% C.L. according to our analysis. These parameters were
chosen in a way to not change the total number of predicted events, so we can
focus on the energy spectrum information. It is clear, again using a visual
comparison, that these new set of parameters produce a worse fit to the data,
fact that is confirmed by a full statistical analysis.

As it was pointed out earlier, the two main neutrino observables that we are
taking into consideration are the neutrino energy and the time of detection.
After discussing the first on the above analysis, we will now focus on the
second, and the best way to do this is the limits in neutrino mass that can be
achieved using this statistical method.

##  6 Neutrino mass limits

An important remark is that the neutrino detection spread in time is an
important source of physical information, allowing us to probe both Supernova
explosion mechanisms and neutrino properties. The most important neutrino
property that can be probed by such time spread is its mass.

The first difficulty in these kind of analysis is that the data itself does
not allow us to correlate the time of arrival of the neutrino burst at the
detectors with the unknown time at which the neutrinos left the Supernova. The
solution is to use the data itself to establish, through statistical analysis,
the match between the neutrino flux theoretical prediction and the data,
taking the time of arrival of the first neutrino event in each detector as a
marker. The time of the following events, $t_{i}$ are taken as relative ones
to the time of arrival of the first event, $t_{1}$:

$\delta_{i}=t_{i}-t_{1}$

and $t_{1}$ is left to vary freely to best match the theoretical prediction in
a previously established time scale.

This simple picture arises when we assume massless neutrinos. In this case the
relative time between events is identical to the relative time between the
emission of these detected neutrinos on the production site, since the time
delay due to the travel between the supernova and the detectors does not
depend on the neutrino properties. In Fig. 3 it is assumed a vanishing mass
neutrino, and the time showed on the animation correspond to the time since
the supernova offset.

Figure 5: Effect of neutrino mass delay on SN1987A detected burst compared to
standard flux for a $3\sigma$ excluded neutrino mass. The gray line
corresponds to the fitted theory in Fig. 3. See the animation at:
https://github.com/santosmv/Animations-visualizing-SN1987A-data-
analysis/blob/main/events_rate_mass_delay.png.

However, since the neutrinos have mass, neutrinos with different energies have
different velocities, which changes the described scenario. More energetic
neutrinos travels faster then less energetic neutrinos, meaning that the
relative times between events does not correspond to relative times of the
neutrinos emission. The correction is done by a simple kinematic analysis:

$t_{i.d}=t_{i,p}+\frac{D}{v_{i}}=t_{i,p}+\frac{D}{c}\sqrt{1-\frac{m^{2}}{p_{i}^{2}}}\sim
t_{i,p}+\frac{D}{c}\left(1-\frac{m^{2}}{2E_{i}^{2}}\right)$

where $D$ is the distance to the supernova, and $m$ and $E$ are the neutrino
mass and event energy. The sub-index $p$ ($d$) refers to the time at
production (detection). The emission time of each event is then calculated
from the relative times $\delta_{i}$, and the kinematic corrections:

$\displaystyle t_{i,d}$ $\displaystyle=$ $\displaystyle t_{1,d}+\delta_{i}$
$\displaystyle t_{i,p}$ $\displaystyle=$
$\displaystyle\delta_{i}+\left(t_{i,p}-\frac{D}{c}\frac{m^{2}}{2E_{1}^{2}}\right)+\frac{D}{c}\frac{m^{2}}{2E_{i}^{2}}$
(10)

For more details, we refere to [8, 10].

Instead of making the correction on the time of the production, presented here
to give proper credit to the authors that proposed and performed this
analysis, we prefer to correct the theoretical predictions by continuous
spread in time on the neutrino flux spectrum at the detector. So, instead of
converting the time of the detected events to the supernova emission, we
adjust the theoretical prediction to the detector site. Clearly both choices
are equivalent, but with this second procedure we can use the same data
animation presented in Fig. 3, and adjust the theoretical curve by making the
replacement:

$t\rightarrow t-\frac{D}{c}\frac{m^{2}}{2E^{2}}$

in Eq. (6).

An animation evidencing this model dependent limit is shown in Fig. 5, for a
$30$ eV neutrino mass, and the same astrophysical parameters used to produce
Fig. 3, and then with the same neutrino flux at the source. But due to the
different time lag of neutrinos traveling to Earth with different energies,
the time history of the expected number of events changes significantly,
allowing us to place a limit on neutrino analysis using a proper statistical
analysis.

##  7 Conclusion

This paper intended to present a pedagogical view of how to understand the
likelihood analysis when an event-by-event treatment is necessary. The
detection of SN1987A is a perfect example for that, once a lot of physics can
be extracted by the few events that were collected through neutrino detection.
It also has the interesting feature that different information can be
extracted from the total expected number of events, its spectral distortion or
its time structure. We present some animations as a visual tool to understand
the statistical procedure, and produce a first impression on how different
models fit the data.

## Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The authors are
also thankful for the support of FAPESP funding Grant 2014/19164-6. The
authors are thankful to Pedro Dedin for usefull discussions during the
production of this article.

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