# Multiplicative independence in the sequence of
$k-$generalized Lucas numbers

Herbert Batte1,∗, Mahadi Ddamulira1, Juma Kasozi1, and Florian Luca2

###### Abstract

Let $(L_{n}^{(k)})_{n\geq 2-k}$ be the sequence of $k-$generalized Lucas
numbers for some fixed integer $k\geq 2$, whose first $k$ terms are
$0,\;\ldots\;,\;0,\;2,\;1$ and each term afterward is the sum of the preceding
$k$ terms. In this paper, we find all pairs of the $k-$generalized Lucas
numbers that are multiplicatively dependent.

Keywords and phrases: $k-$generalized Lucas numbers; multiplicatively
independent integers; linear forms in logarithms.

2020 Mathematics Subject Classification: 11B39, 11D61, 11D45.

∗ Corresponding author

##  1 Introduction

###  1.1 Background

Let $k\geq 2$ be an integer. The sequence of $k$-generalized Fibonacci numbers
$(F_{n}^{(k)})_{n\in{\mathbb{Z}}}$ is the sequence given by

$F_{n}^{(k)}=F_{n-1}^{(k)}+\cdots+F_{n-k}^{(k)}$

with $F_{1}^{(k)}=1$ and $F_{i}^{(k)}=0$ for $i=2-k,3-k,\ldots,0$. If one
keeps the same recurrence relation (every term is the sum of the previous $k$
terms) but instead of the initial values being $0,0,\ldots,0,0,1$ one starts
with $0,0,\ldots,0,2,1$, then one obtains the sequence of $k-$generalized
Lucas numbers. When $k=2$, these coincide with the classical sequences of
Fibonacci and Lucas numbers. If $a,~{}b$ are nonzero integers then we say that
they are multiplicatively independent if the only solution in integers $x,y$
of the equation $a^{x}b^{y}=1$ is $x=y=0$ and multiplicatively dependent
otherwise. Note that any pair of integers such that one of them is $1$ is a
pair of multiplicatively dependent integers.

In [10], Gómez and Luca fixed $k\geq 2$ and looked at the instances $n>m\geq
1$ for which $F_{n}^{(k)}$ and $F_{m}^{(k)}$ are multiplicatively dependent.
This is certainly so if $m\in\\{1,2\\}$ since $F_{1}^{(k)}=F_{2}^{(k)}=1$. In
addition, since $F_{m}^{(k)}=2^{m-2}$ for all $m\in[2,k+1]$, it follows that
$F_{n}^{(k)}$ and $F_{m}^{(k)}$ are multiplicatively dependent when $3\leq
m<n\leq k+1$. The main result of [10] is that these are the only instances
when $F_{n}^{(k)}$ and $F_{m}^{(k)}$ are multiplicatively dependent when
$k\geq 3$. When $k=2$, there is the additional instance $(m,n)=(3,6)$ as
$F_{3}^{(2)}=2$ and $F_{6}^{(2)}=2^{3}$. The instance $k=2$ can be easily
treated with Carmichael’s primitive divisor theorem [6] which asserts that
$F_{n}^{(2)}$ has a primitive prime factor (a prime factor $p$ which does not
divide $F_{m}^{(2)}$ for any positive integer $m<n$) provided $n>12$. Then an
immediate verification of the cases $3\leq m\leq n\leq 12$ gives the solution
$(m,n)=(3,6)$.

Similar results concerning the multiplicative dependence between terms of
other linear recurrence sequences can be seen in [9] and [2]. In this paper,
we investigate the multiplicative dependence between $L_{m}^{(k)}$ and
$L_{n}^{(k)}$ for $n>m$. We assume again that $n\neq 1,~{}m\neq 1$ as
$L_{1}^{(k)}=1$. We prove the following result.

###  1.2 Main Result

######  Theorem 1.1.

Let $k\geq 2$ and $(L_{n}^{(k)})_{n\in{\mathbb{Z}}}$ be the sequence of
$k$–generalized Lucas numbers. If $n>m\geq 0$ are integers with $n\neq
1,~{}m\neq 1$ such that $L_{n}^{(k)}$ and $L_{m}^{(k)}$ are multiplicatively
dependent then $(k,m,n)=(2,0,3),~{}(3,0,7)$.

##  2 Methods

###  2.1 Preliminaries

The following is a well known identity

$\displaystyle L_{n}^{(k)}=3\cdot 2^{n-2},\qquad\text{for all}\qquad 2\leq
n\leq k.$ (2.1)

It can be easily proved by induction. The characteristic polynomial of
$(L_{n}^{(k)})_{n\in{\mathbb{Z}}}$ is given by

$\Psi_{k}(x)=x^{k}-x^{k-1}-\cdots-x-1.$

This polynomial is irreducible in $\mathbb{Q}[x]$. The polynomial
$\Psi_{k}(x)$ possesses a unique real root $\alpha:=\alpha(k)>1$, see [15].
All the other roots of $\Psi_{k}(x)$ are inside the unit circle, see [16]. The
particular root $\alpha$ can be found in the interval

$\displaystyle 2(1-2^{-k})<\alpha<2,\qquad\text{for}\qquad k\geq 2,$ (2.2)

as noted in [20]. As in the classical case when $k=2$, it was shown in [5]
that

$\displaystyle\alpha^{n-1}\leq L_{n}^{(k)}\leq 2\alpha^{n},\qquad\text{holds
for all}\qquad n\geq 1,\quad k\geq 2.$ (2.3)

For $k\geq 2$ we define

$\displaystyle f_{k}(x):=\dfrac{x-1}{2+(k+1)(x-2)}.$

If $2(1-2^{-k})<x<2$, then $\frac{\partial}{\partial x}f_{k}(x)<0$ and so
inequality (2.2) implies that

$\displaystyle\dfrac{1}{2}=f_{k}(2)<f_{k}(\alpha)<f_{k}(2(1-2^{-k}))\leq\dfrac{3}{4},$
(2.4)

holds for all $k\geq 3$. The inequality $1/2=f_{k}(2)<f_{k}(\alpha)<3/4$ also
holds for $k=2$. In addition, it is easy to verify that
$|f_{k}(\alpha_{i})|<1$ holds for all $2\leq i\leq k$, where $\alpha_{i}$ for
$i=2,\ldots,k$ are the roots of $\Psi_{k}(x)$ inside the unit disk. Lastly, it
was shown in [5] that

$\displaystyle
L_{n}^{(k)}=\displaystyle\sum_{i=1}^{k}(2\alpha_{i}-1)f_{k}(\alpha_{i})\alpha_{i}^{n-1}~{}~{}\text{and}~{}~{}\left|L_{n}^{(k)}-f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}\right|<\dfrac{3}{2},$
(2.5)

hold for all $k\geq 2$ and $n\geq 2-k$. This means that
$L_{n}^{(k)}=f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}+e_{k}(n)$, where
$|e_{k}(n)|<1.5$. The first expression in (2.5) is known as the Binet-like
formula for $L_{n}^{(k)}$. Furthermore, the second inequality expression in
(2.5) shows that the contribution of the zeros that are inside the unit circle
to $L_{n}^{(k)}$ is small.

###  2.2 Linear forms in logarithms

We use Baker-type lower bounds for nonzero linear forms in logarithms of
algebraic numbers. There are many such bounds mentioned in the literature but
we use one of Matveev from [14]. Before we can formulate such inequalities we
need the notion of height of an algebraic number recalled below.

######  Definition 2.1.

Let $\gamma$ be an algebraic number of degree $d$ with minimal primitive
polynomial over the integers

$a_{0}x^{d}+a_{1}x^{d-1}+\cdots+a_{d}=a_{0}\prod_{i=1}^{d}(x-\gamma^{(i)}),$

where the leading coefficient $a_{0}$ is positive. Then, the logarithmic
height of $\gamma$ is given by

$h(\gamma):=\dfrac{1}{d}\Big{(}\log
a_{0}+\sum_{i=1}^{d}\log\max\\{|\gamma^{(i)}|,1\\}\Big{)}.$

In particular, if $\gamma$ is a rational number represented as $\gamma=p/q$
with coprime integers $p$ and $q\geq 1$, then $h(\gamma)=\log\max\\{|p|,q\\}$.
The following properties of the logarithmic height function $h(\cdot)$ will be
used in the rest of the paper without further reference:

$\displaystyle h(\gamma_{1}\pm\gamma_{2})$ $\displaystyle\leq
h(\gamma_{1})+h(\gamma_{2})+\log 2;$ $\displaystyle
h(\gamma_{1}\gamma_{2}^{\pm 1})$ $\displaystyle\leq
h(\gamma_{1})+h(\gamma_{2});$ $\displaystyle h(\gamma^{s})$
$\displaystyle=|s|h(\gamma)\quad{\text{\rm valid for}}\quad s\in\mathbb{Z}.$

With these properties, it was easily computed in Section 3 of [10] that

$\displaystyle h\left(f_{k}(\alpha)\right)<2\log k,\qquad\text{for all}\qquad
k\geq 2.$ (2.6)

A linear form in logarithms is an expression of the form

$\Lambda:=b_{1}\log\gamma_{1}+\cdots+b_{t}\log\gamma_{t},$ (2.7)

where $\gamma_{1},\ldots,\gamma_{t}$ are positive real algebraic numbers and
$b_{1},\ldots,b_{t}$ are nonzero integers. We assume, $\Lambda\neq 0$. We need
lower bounds for $|\Lambda|$. We write
${\mathbb{K}}={\mathbb{Q}}(\gamma_{1},\ldots,\gamma_{t})$ and $D$ for the
degree of ${\mathbb{K}}$. We give the general form due to Matveev [14].

######  Theorem 2.1 (Matveev, [14]).

Put $\Gamma:=\gamma_{1}^{b_{1}}\cdots\gamma_{t}^{b_{t}}-1=e^{\Lambda}-1$. Then

$\log|\Gamma|>-1.4\cdot 30^{t+3}\cdot t^{4.5}\cdot D^{2}(1+\log D)(1+\log
B)A_{1}\cdots A_{t},$

where $B\geq\max\\{|b_{1}|,\ldots,|b_{t}|\\}$ and
$A_{i}\geq\max\\{Dh(\gamma_{i}),|\log\gamma_{i}|,0.16\\}$ for $i=1,\ldots,t$.

In our application of Matveev’s result (Theorem 2.1), we need to ensure that
the linear forms in logarithms are indeed nonzero. To ensure this, we shall
need the following two results given as Lemma 2.7 and Lemma 2.8 in [9].

######  Lemma 2.1 (Lemma 2.7 in [9]).

The only solution in positive integers $k\geq 2$, $x$, $y$ of the Diophantine
equation

$(2^{k+1}-3)^{x}=\left(\frac{2^{k+1}k^{k}-(k+1)^{k+1}}{(k-1)^{2}}\right)^{y}$

is $k=2$, for which $x=y$.

######  Lemma 2.2 (Lemma 2.8 in [9]).

Let $N:=N_{\mathbb{K}/\mathbb{Q}},$ where $\mathbb{K}=\mathbb{Q}(\alpha)$.
Then

  1. (i)

For $n,m\geq 1$ and $k\geq 2$, $|N(\alpha)|=1$.

  2. (ii)

$N(2\alpha-1)=2^{k+1}-3$ and
$N(f_{k}(\alpha))=(k-1)^{2}/(2^{k+1}k^{k}-(k+1)^{k+1})$.

  3. (iii)

For $k\geq 3$, $N((2\alpha-1)f_{k}(\alpha))<1$.

For the special case when $t=2$ in Theorem 2.1, we instead apply a sharper
result from [12]. Suppose that $A_{1}>1,~{}A_{2}>1$ are real numbers such that

$\log
A_{i}\geq\max\left\\{h(\gamma_{i}),\frac{|\log\gamma_{i}|}{D},\frac{1}{D}\right\\}\quad{\text{\rm
for}}\quad i=1,2.$ (2.8)

Put

$b^{\prime}:=\frac{|b_{1}|}{D\log A_{2}}+\frac{|b_{2}|}{D\log A_{1}}.$

The following result is Corollary 2 in [12].

######  Theorem 2.2 (Laurent et al, 1995).

In case $t=2$, we have

$\log|\Lambda|\geq-24.34D^{4}\left(\max\left\\{\log
b^{\prime}+0.14,\frac{21}{D},\frac{1}{2}\right\\}\right)^{2}\log A_{1}\log
A_{2}.$

However, during the calculations, upper bounds on the variables which are too
large are obtained, thus there is need to reduce them. In this paper we use
the following result related with continued fractions (see Theorem 8.2.4 in
[17]).

######  Lemma 2.3 (Legendre).

Let $\tau$ be an irrational number, $[a_{0},a_{1},a_{2},\ldots]$ be the
continued fraction expansion of $\tau$. Let
$p_{i}/q_{i}=[a_{0},a_{1},a_{2},\ldots,a_{i}]$, for all $i\geq 0$, be all the
convergents of the continued fraction of $\tau$, and $M$ be a positive
integer. Let $N$ be a non-negative integer such that $q_{N}>M$. Then putting
$a(M):=\max\\{a_{i}:i=0,1,2,\ldots,N\\}$, the inequality

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

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

However, since there are no methods based on continued fractions to find a
lower bound for linear forms in more than two variables with bounded integer
coefficients, we use at some point a method based on the LLL–algorithm. We
next explain this method.

Let $k$ be a positive integer. A subset $\mathcal{L}$ of the $k$–dimensional
real vector space ${\mathbb{R}^{k}}$ is called a lattice if there exists a
basis $\\{b_{1},b_{2},\ldots,b_{k}\\}$ of $\mathbb{R}^{k}$ such that

$\displaystyle\mathcal{L}=\sum_{i=1}^{k}\mathbb{Z}b_{i}=\left\\{\sum_{i=1}^{k}r_{i}b_{i}\mid
r_{i}\in\mathbb{Z}\right\\}.$

We say that $b_{1},b_{2},\ldots,b_{k}$ form a basis for $\mathcal{L}$, or that
they span $\mathcal{L}$. We call $k$ the rank of $\mathcal{L}$. The
determinant $\text{det}(\mathcal{L})$, of $\mathcal{L}$ is defined by

$\displaystyle\text{det}(\mathcal{L})=|\det(b_{1},b_{2},\ldots,b_{k})|,$

with the $b_{i}$’s being written as column vectors. This is a positive real
number that does not depend on the choice of the basis (see [7], Section 1.2).

Given linearly independent vectors $b_{1},b_{2},\ldots,b_{k}$ in
$\mathbb{R}^{k}$, we refer back to the Gram–Schmidt orthogonalization
technique. This method allows us to inductively define vectors $b^{*}_{i}$
(with $1\leq i\leq k$) and real coefficients $\mu_{i,j}$ (for $1\leq j\leq
i\leq k$). Specifically,

$\displaystyle b^{*}_{i}$
$\displaystyle=b_{i}-\sum_{j=1}^{i-1}\mu_{i,j}b^{*}_{j},~{}~{}~{}\mu_{i,j}=\dfrac{\langle
b_{i},b^{*}_{j}\rangle}{\langle b^{*}_{j},b^{*}_{j}\rangle},$

where $\langle\cdot,\cdot\rangle$ denotes the ordinary inner product on
$\mathbb{R}^{k}$. Notice that $b^{*}_{i}$ is the orthogonal projection of
$b_{i}$ on the orthogonal complement of the span of $b_{1},\ldots,b_{i-1}$,
and that $\mathbb{R}b_{i}$ is orthogonal to the span of
$b^{*}_{1},\ldots,b^{*}_{i-1}$ for $1\leq i\leq k$. It follows that
$b^{*}_{1},b^{*}_{2},\ldots,b^{*}_{k}$ is an orthogonal basis of
$\mathbb{R}^{k}$.

######  Definition 2.2.

The basis $b_{1},b_{2},\ldots,b_{n}$ for the lattice $\mathcal{L}$ is called
reduced if

$\displaystyle\|\mu_{i,j}\|$
$\displaystyle\leq\frac{1}{2},\quad\text{for}\quad 1\leq j<i\leq
n,~{}~{}\text{and}$ $\displaystyle\|b^{*}_{i}+\mu_{i,i-1}b^{*}_{i-1}\|^{2}$
$\displaystyle\geq\frac{3}{4}\|b^{*}_{i-1}\|^{2},\quad\text{for}\quad 1<i\leq
n,$

where $\|\cdot\|$ denotes the ordinary Euclidean length. The constant
${3}/{4}$ above is arbitrarily chosen, and may be replaced by any fixed real
number $y$ in the interval ${1}/{4}<y<1$ (see [13], Section 1).

Let $\mathcal{L}\subseteq\mathbb{R}^{k}$ be a $k-$dimensional lattice with
reduced basis $b_{1},\ldots,b_{k}$ and denote by $B$ the matrix with columns
$b_{1},\ldots,b_{k}$. We define

$l\left(\mathcal{L},y\right)=\left\\{\begin{array}[]{c}\min_{x\in\mathcal{L}}||x-y||\quad;~{}~{}y\not\in\mathcal{L}\\\
\min_{0\neq
x\in\mathcal{L}}||x||\quad;~{}~{}y\in\mathcal{L}\end{array}\right.,$

where $||\cdot||$ denotes the Euclidean norm on $\mathbb{R}^{k}$. It is well
known that, by applying the LLL–algorithm, it is possible to give in
polynomial time a lower bound for $l\left(\mathcal{L},y\right)$, namely a
positive constant $c_{1}$ such that $l\left(\mathcal{L},y\right)\geq c_{1}$
holds (see [19], Section V.4).

######  Lemma 2.4.

Let $y\in\mathbb{R}^{k}$ and $z=B^{-1}y$ with $z=(z_{1},\ldots,z_{k})^{T}$.
Furthermore,

  1. (i)

if $y\not\in\mathcal{L}$, let $i_{0}$ be the largest index such that
$z_{i_{0}}\neq 0$ and put $\lambda:=\\{z_{i_{0}}\\}$, where $\\{\cdot\\}$
denotes the distance to the nearest integer.

  2. (ii)

if $y\in\mathcal{L}$, put $\lambda:=1$.

Finally, we set

$c_{1}:=\max\limits_{1\leq j\leq
k}\left\\{\dfrac{||b_{1}||}{||b_{j}^{*}||}\right\\}\qquad\text{and}\qquad\delta:=\lambda\dfrac{||b_{1}||}{c_{1}}.$

In our application, we are given real numbers
$\eta_{0},\eta_{1},\ldots,\eta_{k}$ which are linearly independent over
$\mathbb{Q}$ and two positive constants $c_{3}$ and $c_{4}$ such that

$\displaystyle|\eta_{0}+x_{1}\eta_{1}+\cdots+x_{k}\eta_{k}|\leq
c_{3}\exp(-c_{4}H),$ (2.9)

where the integers $x_{i}$ are bounded as $|x_{i}|\leq X_{i}$ with $X_{i}$
given upper bounds for $1\leq i\leq k$. We write $X_{0}:=\max\limits_{1\leq
i\leq k}\\{X_{i}\\}$. The basic idea in such a situation, from [8], is to
approximate the linear form (2.9) by an approximation lattice. So, we consider
the lattice $\mathcal{L}$ generated by the columns of the matrix

$\mathcal{A}=\begin{pmatrix}1&0&\ldots&0&0\\\ 0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&0&\ldots&1&0\\\ \lfloor
C\eta_{1}\rfloor&\lfloor C\eta_{2}\rfloor&\ldots&\lfloor
C\eta_{k-1}\rfloor&\lfloor C\eta_{k}\rfloor\end{pmatrix},$

where $C$ is a large constant usually of the size of about $X_{0}^{k}$ . Let
us assume that we have an LLL–reduced basis $b_{1},\ldots,b_{k}$ of
$\mathcal{L}$ and that we have a lower bound $l\left(\mathcal{L},y\right)\geq
c_{1}$ with $y:=(0,0,\ldots,-\lfloor C\eta_{0}\rfloor)$. Note that $c_{1}$ can
be computed by using the results of Lemma 2.4. Then, with these notations the
following result is Lemma VI.1 in [19].

######  Lemma 2.5 (Lemma VI.1 in [19]).

Let $S:=\displaystyle\sum_{i=1}^{k-1}X_{i}^{2}$ and
$T:=\dfrac{1+\sum_{i=1}^{k}X_{i}}{2}$. If $\delta^{2}\geq T^{2}+S$, then
inequality (2.9) implies that we either have $x_{1}=x_{2}=\cdots=x_{k-1}=0$
and $x_{k}=-\dfrac{\lfloor C\eta_{0}\rfloor}{\lfloor C\eta_{k}\rfloor}$, or

$H\leq\dfrac{1}{c_{4}}\left(\log(Cc_{3})-\log\left(\sqrt{\delta^{2}-S}-T\right)\right).$

Finally, we present an analytic argument which is Lemma 7 in [11].

######  Lemma 2.6 (Lemma 7 in [11]).

If $s\geq 1$, $T>(4s^{2})^{s}$ and $T>\displaystyle\frac{z}{(\log z)^{s}}$,
then

$z<2^{s}T(\log T)^{s}.$

SageMath 9.5 is used to perform all computations in this work.

##  3 Proof of the main result

We do this in the following cases.

###  3.1 The case $k=2$

In this subsection we study the Diophantine equation

$(L_{n})^{x}=(L_{m})^{y},$ (3.1)

where $n>m\geq 0$, $n\neq 1,m\neq 1$ and $x,y\in\mathbb{Z}$.

######  Lemma 3.1.

The only integer solution $(n,m)$ of equation (3.1) in the range $n>m\geq 0$,
$n\neq 1,m\neq 1$ is $(n,m)=(3,0)$.

###### Proof.

By Carmichael’s Primitive Divisor Theorem and the known relation
$F_{2k}=F_{k}L_{k}$, it follows that for $k>6$, $L_{k}$ has a primitive prime
factor (namely, the primitive prime factor of $F_{2k}$). In particular,
equation (3.1) is impossible with $x\neq 0$ if $n>6$. One now checks that in
the remaining range $0\leq m<n\leq 6$, the only convenient solution is
$(m,n)=(3,0)$. ∎

###  3.2 The case $k\geq 3$

The equation is

$(L_{n}^{(k)})^{x}=(L_{m}^{(k)})^{y},$ (3.2)

where none of $x,~{}y$ is zero. Since $n>m\geq 0$, and $n\neq 1,~{}m\neq 1$ we
have that $n\geq 2$. If $m\in[0,k]$, $m\neq 1$, we then have that either $m=0$
and $L_{0}^{(k)}=2$ or $m\geq 2$ and $L_{m}^{(k)}=3\cdot 2^{m-2}$. From the
above formulas, we get that if $0\leq m\leq k$, $m\neq 1$, then $n\geq k+1$.
Furthermore, $P(L_{n}^{(k)})\leq 3$. All $k$–generalized Lucas numbers
$L_{n}^{(k)}$ with $P(L_{n}^{(k)})\leq 7$ and $n\geq k+1$ have been found in
[1] and they are the following

$\displaystyle
L_{3}^{(2)}=2^{2},~{}~{}~{}~{}L_{4}^{(2)}=7,~{}~{}~{}~{}L_{6}^{(2)}=2\cdot
3^{2},~{}~{}~{}~{}L_{4}^{(3)}=2\cdot 5,~{}~{}~{}~{}L_{6}^{(3)}=5\cdot 7,$
$\displaystyle L_{7}^{(3)}=2^{6},~{}~{}~{}~{}L_{12}^{(3)}=2\cdot 3^{3}\cdot
5^{2},~{}~{}~{}~{}L_{15}^{(3)}=2^{4}\cdot 3\cdot 5^{2}\cdot
7,~{}~{}~{}~{}L_{8}^{(4)}=2^{5}\cdot
5~{}~{}\text{and}~{}~{}L_{15}^{(10)}=2^{2}\cdot 5^{3}\cdot 7^{2}.$

We learn from the above list that the only possibilities are $(k,m,n)=(2,0,3)$
and $(3,0,6)$. From now on, we assume that $P(L_{m}^{(k)})\geq 11$. In
particular, $n>m\geq k+1$. In addition, $L_{n}^{(k)}>L_{m}^{(k)}$ showing that
$x<y$. It is a straightforward exercise to show that relation (2.3) together
with (3.2) imply that $x<m$ and $y<n$. Therefore, $n=\max\\{n,m,x,y,k\\}$.

Assume next that $n\leq 50$. Then $4\leq n\leq 50$, $3\leq m\leq 49$ and
$3\leq k\leq 48$. We wrote a code in SageMath to find all coincidences for
which equations (3.2) hold. We find none in these ranges, see Appendix 1. From
now on, $n>50$.

###  3.3 An inequality for $n$ in terms of $k$

######  Lemma 3.2.

Let $(n,m,x,y,k)$ be a solution to (3.1) with $n>m\geq k+1$, $k\geq 3$, $n>50$
and $x<y<n$. Then

$\displaystyle n<8.5\cdot 10^{34}k^{8}(\log k)^{6}.$

###### Proof.

Here, we first go back to (2.5) and write it as

$\displaystyle
L_{n}^{(k)}=f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}+e_{k}(n),~{}~{}\text{where}~{}~{}|e_{k}(n)|<1.5.$
(3.3)

Hence,

$\displaystyle\left(L_{n}^{(k)}\right)^{x}=\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}\left[1+\dfrac{e_{k}(n)}{f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}}\right]^{x}.$

Now, we study the elements

$\displaystyle(1+r)^{x}~{}~{}\text{and}~{}~{}z:=xr,~{}~{}~{}~{}\text{where}~{}~{}r=\dfrac{e_{k}(n)}{f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}}.$
(3.4)

Since $k\geq 3$, then $\alpha\geq 1.75$ by relation (2.2) and so
$2\alpha-1\geq 2.5$. Moreover, $f_{k}(\alpha)>{1}/{2}$ by inequality (2.4).
This means that

$\displaystyle|r|=\left|\dfrac{e_{k}(n)}{f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}}\right|<\dfrac{1.5}{0.5\cdot
2.5\cdot 1.5^{n-1}}=\dfrac{1.2}{1.5^{n-1}}<\dfrac{1}{1.5^{n-2}},$

and

$\displaystyle|z|=x|r|<\dfrac{n}{1.5^{n-2}}.$

In particular, $|z|<10^{-6}$ for all $n>50$. Now, if $r<0$, then

$\displaystyle
1>(1+r)^{x}=\exp\left(x\log(1-|r|)\right)\geq\exp(-2|z|)>1-2|z|,$

and if $r>0$, then

$\displaystyle
1<(1+r)^{x}=\left(1+\dfrac{|z|}{x}\right)^{x}<\exp(|z|)<1+2|z|,$

because $|r|<0.5$ for all $n>50$ and $|z|<10^{-6}$ is very small. Therefore in
both cases we have that

$\displaystyle\left|\left(L_{n}^{(k)}\right)^{x}-\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}\right|$
$\displaystyle<2|z|\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}$
$\displaystyle<\dfrac{2n\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}}{1.5^{n-2}}.$
(3.5)

We now use equation (3.2) together with (3.3) to have

$\displaystyle\left|\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}-\left(f_{k}(\alpha)\right)^{y}(2\alpha-1)^{y}\alpha^{(m-1)y}\right|$
$\displaystyle=\left|\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}-\left(f_{k}(\alpha)\right)^{y}(2\alpha-1)^{y}\alpha^{(m-1)y}-\left(L_{n}^{(k)}\right)^{x}+\left(L_{m}^{(k)}\right)^{y}\right|$
$\displaystyle<\left|\left(L_{n}^{(k)}\right)^{x}-\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}\right|+\left|\left(L_{m}^{(k)}\right)^{y}-\left(f_{k}(\alpha)\right)^{y}(2\alpha-1)^{y}\alpha^{(m-1)y}\right|$
$\displaystyle<\dfrac{2n\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}}{1.5^{n-2}}+\dfrac{2m\left(f_{k}(\alpha)\right)^{y}(2\alpha-1)^{y}\alpha^{(m-1)y}}{1.5^{m-2}}$
$\displaystyle<\dfrac{4m}{1.5^{m-2}}\left(\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}+\left(f_{k}(\alpha)\right)^{y}(2\alpha-1)^{y}\alpha^{(m-1)y}\right),$
(3.6)

where in the last inequality we have used the fact that $x\mapsto x/1.5^{x}$
is decreasing for all $x\geq 3$, together with $n>m\geq k+1\geq 4$.

Dividing both sides of (3.3) by
$\max\left\\{\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x},~{}\left(f_{k}(\alpha)\right)^{y}(2\alpha-1)^{y}\alpha^{(m-1)y}\right\\}$,
we get

$\displaystyle\left|1-\left(f_{k}(\alpha)\right)^{\Delta(y-x)}(2\alpha-1)^{\Delta(y-x)}\alpha^{\Delta((m-1)y-(n-1)x)}\right|$
$\displaystyle<\dfrac{4m}{1.5^{m-2}}\left(1+\left(f_{k}(\alpha)\right)^{\Delta(y-x)}(2\alpha-1)^{\Delta(y-x)}\alpha^{\Delta((m-1)y-(n-1)x)}\right)<\dfrac{4m}{1.5^{m-2}}(1+1)<\dfrac{19m}{1.5^{m}},$
(3.7)

where $\Delta\in\\{\pm 1\\}$. We now intend to apply Theorem 2.1 on the
left–hand side of (3.3). Let

$\Gamma:=\left(f_{k}(\alpha)\right)^{\Delta(y-x)}(2\alpha-1)^{\Delta(y-x)}\alpha^{\Delta((m-1)y-(n-1)x)}-1=e^{\Lambda}-1.$

If $\Lambda=0$, then

$\displaystyle\left(f_{k}(\alpha)\right)^{y-x}=(2\alpha-1)^{x-y}\alpha^{(n-1)x-(m-1)y}.$

Applying norms in ${\mathbb{K}}={\mathbb{Q}}(\alpha)$ and using
$|N(\alpha)|=1$, and item (i) of Lemma 2.2, the above equation becomes

$\displaystyle|N\left(f_{k}(\alpha)\right)|^{y-x}=|N(2\alpha-1)|^{x-y},$

which implies that

$\displaystyle\dfrac{1}{|N\left(f_{k}(\alpha)\right)|}=|N(2\alpha-1)|.$

Here, we used that $y>x$ (in particular, $y-x\neq 0$). Now, by item (ii) of
Lemma 2.2, we have from the above that

$\frac{2^{k+1}k^{k}-(k+1)^{k+1}}{(k-1)^{2}}=2^{k+1}-3,$

which gives $k=2$, by Lemma 2.1. This contradicts the working assumption that
$k\geq 3$. Thus, $\Lambda\neq 0$. We use the field $\mathbb{Q}(\alpha)$ with
degree $D=k$. Here, $t=3$,

$\displaystyle\gamma_{1}$
$\displaystyle:=f_{k}(\alpha),~{}~{}~{}~{}~{}~{}\gamma_{2}:=2\alpha-1,~{}~{}~{}~{}~{}\gamma_{3}:=\alpha,$
$\displaystyle b_{1}$
$\displaystyle:=\Delta(y-x),~{}~{}b_{2}:=\Delta(y-x),~{}~{}b_{3}:=\Delta((m-1)y-(n-1)x).$

We need to ensure that $B\geq\max\\{|b_{1}|,|b_{2}|,|b_{3}|\\}$, so we can
take $B:=n^{2}$. We also need to ensure that
$A_{i}\geq\max\\{Dh(\gamma_{i}),|\log\gamma_{i}|,0.16\\}$ for $i=1,2,3$. So,
we take $A_{1}:=Dh(\gamma_{1})=2k\log k$. Since $h(\alpha)=(\log\alpha)/k$ and
$h(2\alpha-1)<3/k$, we can take $A_{2}:=A_{3}:=1$. Then, by Theorem 2.1,

$\displaystyle\log|\Gamma|$ $\displaystyle>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot
k^{2}(1+\log k)(1+2\log n)\cdot 2k\log k$ $\displaystyle>-1.6\cdot
10^{12}k^{3}(\log k)^{2}\log n,$ (3.8)

since $k\geq 3$ and $n>50$. Comparing (3.3) and (3.3), we get

$\displaystyle m\log 1.5-\log 19m$ $\displaystyle<1.6\cdot 10^{12}k^{3}(\log
k)^{2}\log n$ (3.9)

Next, we proceed in two cases, that is, we distinguish between the cases
$n\leq m^{2}$ and $n>m^{2}$.

  1. (a)

When $n\leq m^{2}$, we use (3.9) to obtain that

$\displaystyle m\log 1.5-\log 19m$ $\displaystyle<1.6\cdot 10^{12}k^{3}(\log
k)^{2}\log m^{2};$ $\displaystyle m\log 1.5$ $\displaystyle<3.2\cdot
10^{12}k^{3}(\log k)^{2}\log m+\log 19+\log m;$ $\displaystyle\dfrac{m}{\log
m}$ $\displaystyle<8\cdot 10^{12}k^{3}(\log k)^{2}$

We now apply Lemma 2.6 with the data $z:=m$, $s:=1$ and $T:=8\cdot
10^{12}k^{3}(\log k)^{2}>(4s^{2})^{s}=4$ since $k\geq 3$. We get

$\displaystyle m$ $\displaystyle<2\cdot 8\cdot 10^{12}k^{3}(\log
k)^{2}\log[8\cdot 10^{12}k^{3}(\log k)^{2}]$ $\displaystyle=1.6\cdot
10^{13}k^{3}(\log k)^{2}\left(\log 8\cdot 10^{12}+3\log k+2\log\log k\right)$
$\displaystyle<1.6\cdot 10^{13}k^{3}(\log k)^{2}\left(30+5\log
k\right)<5.3\cdot 10^{14}k^{3}(\log k)^{3}.$ (3.10)

So, we get that $n\leq m^{2}<\left(5.3\cdot 10^{14}k^{3}(\log
k)^{3}\right)^{2}<2.9\times 10^{29}k^{6}(\log k)^{6}$.

  2. (b)

When $n>m^{2}$, we return to (3.2) and rewrite it using (3.3) as

$\displaystyle\left|\left(L_{m}^{(k)}\right)^{y}-\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}\right|$
$\displaystyle<\dfrac{2n\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}}{1.5^{n-2}}.$

Dividing both sides of the above inequality by
$\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}$, we get

$\displaystyle\left|\left(f_{k}(\alpha)\right)^{-x}(2\alpha-1)^{-x}\alpha^{-(n-1)x}\left(L_{m}^{(k)}\right)^{y}-1\right|$
$\displaystyle<\dfrac{4.5n}{1.5^{n}}<\dfrac{1}{1.5^{n/2}},$ (3.11)

since $n>50$. We again apply Theorem 2.1 on the left–hand side of (3.11).
Here, we let

$\Gamma_{1}:=\left(f_{k}(\alpha)\right)^{-x}(2\alpha-1)^{-x}\alpha^{-(n-1)x}\left(L_{m}^{(k)}\right)^{y}-1=e^{\Lambda_{1}}-1.$

Again, $\Lambda_{1}\neq 0$, otherwise we would have

$\left(f_{k}(\alpha)\right)^{x}(2\alpha-1)^{x}\alpha^{(n-1)x}=\left(L_{m}^{(k)}\right)^{y}=\left(L_{n}^{(k)}\right)^{x}.$

Upon simplifying the exponents, we get
$f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}=L_{n}^{(k)}$. Taking norms in
${\mathbb{K}}={\mathbb{Q}}(\alpha)$ and using that $|N(\alpha)|=1$, by item
(i) and (ii) of Lemma 2.2, we get that

$\displaystyle
1>N((2\alpha-1)f_{k}(\alpha))=N(L_{n}^{(k)})=(L_{n}^{(k)})^{k}>1,$

a contradiction. Thus, $\Lambda_{1}\neq 0$. We again use the field
$\mathbb{Q}(\alpha)$ with degree $D=k$. Here, $t=4$,

$\displaystyle\gamma_{1}$
$\displaystyle:=f_{k}(\alpha),~{}~{}\gamma_{2}:=2\alpha-1,~{}~{}\gamma_{3}:=\alpha,~{}~{}\gamma_{4}:=L_{m}^{(k)},$
$\displaystyle b_{1}$
$\displaystyle:=-x,~{}~{}b_{2}:=-x,~{}~{}b_{3}:=-(n-1)x~{}~{}b_{4}=y.$

We need $B\geq\max\\{|b_{1}|,|b_{2}|,|b_{3}|,|b_{4}|\\}$, so we can take
$B:=n^{2}$. Like before, we take $A_{1}:=2k\log k$, and we can still take
$A_{2}:=A_{3}:=1$. For $A_{4}$, we note that $L_{m}^{(k)}\leq 2\alpha^{m}$ by
inequality (2.3), so we can take $A_{4}=2mk$. Thus, by Theorem 2.1,

$\displaystyle\log|\Gamma|$ $\displaystyle>-1.4\cdot 30^{7}\cdot 4^{4.5}\cdot
k^{2}(1+\log k)(1+2\log n)\cdot 2k\log k\cdot 2mk$ $\displaystyle>-3.5\cdot
10^{14}k^{4}(\log k)^{2}\sqrt{n}\log
n,~{}~{}~{}\text{since}~{}m^{2}<n~{}~{}\text{in this case (b)}.$ (3.12)

Comparing (3.11) and (b), we get

$\displaystyle\dfrac{n}{2}\log 1.5$ $\displaystyle<3.5\cdot 10^{14}k^{4}(\log
k)^{2}\sqrt{n}\log n;$ $\displaystyle\sqrt{n}$ $\displaystyle<1.8\cdot
10^{15}k^{4}(\log k)^{2}\log n;$ $\displaystyle\dfrac{\sqrt{n}}{\log\sqrt{n}}$
$\displaystyle<3.6\cdot 10^{15}k^{4}(\log k)^{2}.$ (3.13)

Again, we now apply Lemma 2.6 with $z:=\sqrt{n}$, $s:=1$ and $T:=3.6\cdot
10^{15}k^{4}(\log k)^{2}>(4s^{2})^{s}=4$ since $k\geq 3$. We get

$\displaystyle\sqrt{n}$ $\displaystyle<2\times 3.6\cdot 10^{15}k^{4}(\log
k)^{2}\log\left(3.6\cdot 10^{15}k^{4}(\log k)^{2}\right)$
$\displaystyle=7.2\cdot 10^{15}k^{4}(\log k)^{2}\left(\log 3.6\cdot
10^{15}+4\log k+2\log\log k\right)$ $\displaystyle<7.2\cdot 10^{15}k^{4}(\log
k)^{2}\left(36+6\log k\right)$ $\displaystyle<2.9\cdot 10^{17}k^{4}(\log
k)^{3};$ $\displaystyle\text{or}~{}~{}~{}n$ $\displaystyle<8.5\cdot
10^{34}k^{8}(\log k)^{6}.$ (3.14)

This completes the proof of Lemma 3.2. ∎

###  3.4 Considerations on $k$

In our analysis, we consider two distinct scenarios based on the value of $k$:
when $k>1000$ and when $k\leq 1000$. For the first scenario, we show that
equation (3.2) has no solutions. Consequently, we focus on the case where
$k\leq 1000$. However, it is noteworthy that Lemma 3.2 imposes a different
upper bound on $n$, specifically $n<9.3\times 10^{63}$, which is exceedingly
large and impractical for direct computation. To overcome this computational
challenge, we reduce this upper bound in two cases, using inequalities
analogous to (3.3) and (3.11), involving the LLL–algorithm, as we shall
elaborate.

####  3.4.1 The case $k>1000$

For this case, we have from Lemma 3.2 that

$\displaystyle n<8.5\cdot 10^{34}k^{8}(\log k)^{6}<2^{2k/5},$

for all $k>1000$. Next, let $a$ and $b$ be nonnegative integers with $b>0$. We
quote the following inequality which is proved as estimate (3.14) on page 229
of [9]. It asserts that

$\displaystyle\left|\left(u_{n}^{(k)}\right)^{x}-2^{(n-2)x}(a+b)^{x}\right|<71\cdot\dfrac{2^{(n-2)x}(a+b)^{x}}{2^{2k/5}}.$
(3.15)

For us here, $u_{n}^{(k)}:=L_{n}^{(k)}$, $a=L_{0}^{(k)}=2$ and
$b=L_{1}^{(k)}=1$. Therefore, using inequality (3.15) together with relation
(3.2), we get

$\displaystyle\left|2^{(n-2)x}\cdot 3^{x}-2^{(m-2)y}\cdot 3^{y}\right|$
$\displaystyle=\left|2^{(n-2)x}\cdot 3^{x}-2^{(m-2)y}\cdot
3^{y}+\left(L_{m}^{(k)}\right)^{y}-\left(L_{n}^{(k)}\right)^{x}\right|$
$\displaystyle\leq\left|\left(L_{n}^{(k)}\right)^{x}-2^{(n-2)x}\cdot
3^{x}\right|+\left|\left(L_{m}^{(k)}\right)^{y}-2^{(m-2)y}\cdot 3^{y}\right|$
$\displaystyle<71\cdot\dfrac{2^{(n-2)x}\cdot 3^{x}+2^{(m-2)y}\cdot
3^{y}}{2^{2k/5}}.$ (3.16)

If we divide through by $\max\left\\{2^{(n-2)x}\cdot 3^{x},~{}2^{(m-2)y}\cdot
3^{y}\right\\}$, we get

$\displaystyle\left|1-2^{\varepsilon((n-2)x-(m-2)y)}\cdot
3^{\varepsilon(x-y)}\right|<\dfrac{142}{2^{2k/5}},$

where $\varepsilon\in\\{\pm 1\\}$. To proceed, let
$\Gamma_{2}:=1-2^{\varepsilon((n-2)x-(m-2)y)}\cdot 3^{\varepsilon(x-y)}$. If
we write

$\Lambda_{2}:=\varepsilon((n-2)x-(m-2)y)\log 2+\varepsilon(x-y)\log 3,$

then

$\displaystyle|\Lambda_{2}|<e^{|\Lambda_{2}|}\left(e^{\Lambda_{2}}-1\right)<\dfrac{284}{2^{2k/5}},$
(3.17)

since $\left|e^{\Lambda_{2}}-1\right|=|\Gamma_{2}|<142/2^{2k/5}$ and
$e^{|\Lambda_{2}|}\leq 1+|\Gamma_{2}|<2$ for all $k>1000$. We need to ensure
that $\Gamma_{2}\neq 0$. But if $\Gamma_{2}=0$, we then get $x-y=0$ (from the
exponent of $3$) and next $(n-2)x=(m-2)y$ (from the exponent of $2$), so since
$x=y\neq 0$, we get $n-2=m-2$, or $n=m$, a contradiction.

Now, we apply Theorem 2.2 on $\Lambda_{2}$ with $\gamma_{1}:=2$,
$\gamma_{2}:=3$, $b_{1}:=\varepsilon((n-2)x-(m-2)y)$ and
$b_{2}:=\varepsilon(x-y)$. Moreover, $\mathbb{K}:=\mathbb{Q}$, $D=1$ and so we
can write $\log A_{i}:=2$, for $i=1,2$. Also, we have

$b^{\prime}:=\frac{|b_{1}|}{D\log A_{2}}+\frac{|b_{2}|}{D\log
A_{1}}=\dfrac{1}{2}\left(|b_{1}|+|b_{2}|\right).$

Therefore,

$\displaystyle\log|\Lambda_{2}|$
$\displaystyle\geq-24.34\left(\max\left\\{\log
b^{\prime}+0.14,~{}21,~{}0.5\right\\}\right)^{2}\cdot 2\cdot 2$
$\displaystyle>-98\left(\max\left\\{\log
b^{\prime}+0.14,~{}21\right\\}\right)^{2}.$

Using the upper bound from relation (3.17), we get

$\displaystyle\frac{2}{5}k\log 2-\log 284$
$\displaystyle<98\left(\max\left\\{\log
b^{\prime}+0.14,~{}21\right\\}\right)^{2},$ $\displaystyle k$
$\displaystyle<354\left(\max\left\\{\log
b^{\prime}+0.14,~{}21\right\\}\right)^{2}+21.$

Now, if $\max\left\\{\log b^{\prime}+0.14,~{}21\right\\}=21$, then $k<156135$
and if $\max\left\\{\log b^{\prime}+0.14,~{}21\right\\}=\log b^{\prime}+0.14$,
then

$\displaystyle b^{\prime}$
$\displaystyle=\dfrac{1}{2}\left(|b_{1}|+|b_{2}|\right)=\dfrac{1}{2}\left(|((n-2)x-(m-2)y)|+|(x-y)|\right)$
$\displaystyle\leq\dfrac{n+n^{2}}{2}<n^{2}<7.3\cdot 10^{69}k^{16}(\log
k)^{12},$

where we have used Lemma 3.2. Hence,

$\displaystyle\max\left\\{\log b^{\prime}+0.14,~{}21\right\\}$
$\displaystyle=\log b^{\prime}+0.14$ $\displaystyle<\log\left(7.3\cdot
10^{69}k^{16}(\log k)^{12}\right)+0.14$ $\displaystyle=\log(7.3\cdot
10^{69})+16\log k+12\log\log k+0.14$ $\displaystyle<162+28\log k<52\log
k,\quad\text{for}\quad k>1000.$

We thus get that $k<354\cdot(52\log k)^{2}+21<10^{6}(\log k)^{2}$. Applying
Lemma 2.6 with $z:=k$, $s:=2$ and $T:=10^{6}>(4s^{2})^{s}=256$, we get
$k<7.7\cdot 10^{8}$. Therefore, in all cases, $k<7.7\cdot 10^{8}$. With this
upper bound, we have by Lemma 3.2 that $n<7.8\cdot 10^{113}$.

Next, we divide (3.17) by $(x-y)\log 2$, we have

$\displaystyle\left|\dfrac{\log 3}{\log
2}-\dfrac{(n-2)x-(m-2)y}{x-y}\right|<\dfrac{410}{2^{2k/5}(x-y)},$ (3.18)

since $x$ and $y$ are distinct. By Lemma 2.3 with $\tau:=\dfrac{\log 3}{\log
2}$ and $M:=10^{114}$, we have

$\displaystyle\dfrac{1}{(a(M)+2)(x-y)^{2}}<\left|\dfrac{\log 3}{\log
2}-\dfrac{(n-2)x-(m-2)y}{x-y}\right|<\dfrac{410}{2^{2k/5}(x-y)},$

where $a(M)=100$ (in fact, $q_{229}>10^{114}$ and $\max\\{a_{k}:0\leq k\leq
229\\}=a_{218}=100$). The above inequality gives

$\displaystyle\dfrac{1}{(100+2)(x-y)^{2}}$
$\displaystyle<\dfrac{410}{2^{2k/5}(x-y)};$ $\displaystyle 2^{2k/5}$
$\displaystyle<41820(x-y)<41820n;$ $\displaystyle<3.6\cdot 10^{39}k^{8}(\log
k)^{6}.$

Taking logarithms both sides and simplifying, we get

$\displaystyle\dfrac{2}{5}k\log 2<\log(3.6\cdot 10^{39})+8\log k+6\log\log
k<92+14\log k<28\log k,$

which implies that $k<101\log k$. Applying Lemma 2.6 with $z:=k$, $s:=1$ and
$T:=101>(4s^{2})^{s}=4$, we get $k<933$. This contradicts the working
assumption $k>1000$.

####  3.4.2 The case $k\leq 1000$

In this case, since $k\leq 1000$, then $n<8.5\cdot 10^{34}(1000)^{8}(\log
1000)^{6}<9.3\cdot 10^{63}$, by Lemma 3.2. The purpose here is to reduce this
upper bound on $n$. To do this, we first go back to (3.3) and recall that we
had already obtained

$\Gamma:=\left(f_{k}(\alpha)\right)^{\Delta(y-x)}(2\alpha-1)^{\Delta(y-x)}\alpha^{\Delta((m-1)y-(n-1)x)}-1=e^{\Lambda}-1.$

We already showed that $\Lambda\neq 0$, so $\Gamma\neq 0$. Assume for a moment
that $m>14$, then we have that $\left|e^{\Lambda}-1\right|=|\Gamma|<1$ and
$e^{|\Lambda|}\leq 1+|\Gamma|<2$, implying that

$\displaystyle|(m-1)y-(n-1)x|$
$\displaystyle<\dfrac{1}{\log\alpha}\left|2+|y-x|\log(2\alpha-1)+|y-x||\log
f_{k}(\alpha)|\right|$
$\displaystyle<\dfrac{1}{\log\alpha}(2+2n+n)=\dfrac{n}{\log\alpha}\cdot\left(3+\dfrac{2}{n}\right)<9n,$

since $m>14$. In the calculation above, we have used $|\log(2\alpha-1)|<2$,
$|\log f_{k}(\alpha)|<1$ and $n:=\max\\{x,y\\}$. We can therefore rewrite
(3.3) as

$\displaystyle|\Lambda|<e^{|\Lambda|}|e^{\Lambda}-1|<\dfrac{38m}{1.5^{m}}<\dfrac{10^{28}}{1.5^{m}},$

where we have used (a) and $k\leq 1000$.

Now, for each $k\in[3,1000]$, we use the LLL–algorithm to compute a lower
bound for the smallest nonzero number of the form

$|\Lambda|:=\left|(y-x)\log
f_{k}(\alpha)+(y-x)\log(2\alpha-1)+((m-1)y-(n-1)x)\log\alpha\right|,$

with integer coefficients not exceeding $9n<8.4\cdot 10^{64}$ in absolute
value. Specifically, we consider the approximation lattice

$\mathcal{A}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ \lfloor C\log
f_{k}(\alpha)\rfloor&\lfloor C\log(2\alpha-1)\rfloor&\lfloor
C\log\alpha\rfloor\end{pmatrix},$

with $C:=6\cdot 10^{194}$ and choose $y:=\left(0,0,0\right)$. Now, by Lemma
2.4, we get

$l\left(\mathcal{L},y\right)=|\Lambda|>c_{1}=10^{-68}\qquad\text{and}\qquad\delta=10^{66}.$

So, Lemma 2.5 gives $S=2.22\cdot 10^{130}$ and $T=1.26\cdot 10^{65}$. Since
$\delta^{2}\geq T^{2}+S$, then choosing $c_{3}:=10^{28}$ and $c_{4}:=\log
1.5$, we get $m<891$.

Recall that we are trying to reduce the upper bound on $n$. At this point with
a reduced bound on $m$, we proceed in two cases.

  1. (a)

When $n\leq m^{2}$, we have from the reduced bound on $m$ that
$n<891^{2}=793881$. With this new upper bound for $n$ we repeat the
LLL–algorithm to get a lower bound of $|\Lambda|$, where now the coefficients
of the linear form $|\Lambda|$ are integers not exceeding $9n<7.2\cdot
10^{6}$. With the same approximation lattice and $C:=10^{21}$, we get
$|\Lambda|>10^{-16}$, $S=2.54\cdot 10^{14}$ and $T=1.39\cdot 10^{7}$.
Moreover, choosing $c_{3}:=38m=33858$ and $c_{4}:=\log 1.5$, we obtain that
$m<111$, or similarly $n\leq m^{2}<12321$. Repeating this algorithm twice, we
conclude that $n<1600$.

  2. (b)

When $n>m^{2}$, then we go back to (3.11) where we deduced

$\Gamma_{1}:=\left(f_{k}(\alpha)\right)^{-x}(2\alpha-1)^{-x}\alpha^{-(n-1)x}\left(L_{m}^{(k)}\right)^{y}-1=e^{\Lambda_{1}}-1.$

Clearly, $\Gamma_{1}\neq 0$ since we already showed that $\Lambda_{1}\neq 0$.
If $\Gamma_{1}>0$, then $e^{\Lambda_{1}}-1>0$, so from (3.11) we obtain

$0<\Gamma_{1}<\frac{1}{1.5^{n/2}}.$

If $\Gamma_{1}<0$, then we can note from (3.11) that $1/1.5^{n/2}<1/2$ since
$n>50$. Hence, it follows that $|e^{\Lambda_{1}}-1|<1/2$ which implies
$e^{\Lambda_{1}}<2$. Since $\Gamma_{1}<0$, we obtain that

$0<|\Lambda_{1}|\leq
e^{|\Lambda_{1}|}-1=e^{|\Lambda_{1}|}|e^{\Lambda_{1}}-1|<\frac{2}{1.5^{n/2}}.$

Thus, in all cases, we have

$|\Lambda_{1}|<\frac{2}{1.5^{n/2}}.$ (3.19)

Observe that $|\Lambda_{1}|$ is an expression of the form

$|a_{1}\log f_{k}(\alpha)+a_{2}\log(2\alpha-1)+a_{3}\log\alpha+a_{4}\log
L_{m}^{(k)}|,$

where $a_{1}:=-x$, $a_{2}:=-x$, $a_{3}:=-(n-1)x$, $a_{4}:=y$. From Lemma 3.2,
we have

$\max\\{|a_{i}|:1\leq i\leq 4\\}<n^{2}<10^{128}.$

At this point when $k\in[3,1000]$ and $m\in[3,890]$, we consider the
approximation lattice

$\mathcal{A}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ \lfloor C\log
f_{k}(\alpha)\rfloor&\lfloor C\log(2\alpha-1)\rfloor&\lfloor
C\log\alpha\rfloor&\lfloor C\log L_{m}^{(k)}\rfloor\end{pmatrix},$

with $C:=10^{512}$ and choose $y:=\left(0,0,0,0\right)$. By Lemma 2.4, we get
$|\Lambda_{1}|>c_{1}=10^{-130}$ and $\delta=10^{129}$. So, Lemma 2.5 gives
$S=2\cdot 10^{256}$ and $T=1.5\cdot 10^{128}$. Since $\delta^{2}\geq T^{2}+S$,
then choosing $c_{3}:=2$ and $c_{4}:=\log 1.5$, we get $n/2<2178$. This
implies that $n\leq 4356$.

With this new upper bound for $n$ we repeat the LLL–algorithm once again to
get a lower bound of $|\Lambda_{1}|$, where now the coefficients $a_{i}$ are
integers satisfying

$\max\\{|a_{i}|:1\leq i\leq 4\\}<n^{2}<1.9\cdot 10^{7}.$

With the same approximation lattice and $C:=10^{30}$, we get
$|\Lambda_{1}|>10^{-10}$, $\delta=10^{8}$ $S=1.45\cdot 10^{15}$ and
$T=3.81\cdot 10^{7}$. We then obtain that $n\leq 258$.

So, in all cases $n<1600$. Finally, we do a computational search for solutions
to our Diophantine equation (3.2), when $3\leq k\leq 1000$ and $3<m<n<1600$.
First, for $k$ fixed, we use SageMath to calculate

$M(n,m):=\gcd(L^{(k)}_{n},L^{(k)}_{m})$

and we print the pairs $(n,m)$, with $n,m\in[3,1600]$, such that

$\text{power\textunderscore
mod}(M(n,m),n,L^{(k)}_{n})=0\quad\text{and}\quad\text{power\textunderscore
mod}(M(n,m),m,L^{(k)}_{m})=0.$

Here, $\text{power\textunderscore mod}(A,r,B)$ calculate $A^{r}\pmod{B}$. To
conclude, we verify which of these pairs correspond to solutions of equation
(3.2), for each $k\in[3,1000]$. We find none, see Appendix 3. This program ran
for about 20 hours.

## Acknowledgments

The first author thanks the Eastern Africa Universities Mathematics Programme
(EAUMP) for funding his doctoral studies. The first and last co-authors were
supported in part by Grant #2024-029-NUM from Wits, Johannesburg, South
Africa. The last author worked on this paper during a fellowship at STIAS.
This author thanks STIAS for hospitality and support.

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  * [9] Gómez, C. A., Gómez, J. C., & Luca, F. (2020). Multiplicative dependence between $k-$Fibonacci and $k-$Lucas numbers. Periodica Mathematica Hungarica, 81(2), 217-233.
  * [10] Gómez Ruiz, C. A. & Luca, F. (2016). Multiplicative independence in $k-$generalized Fibonacci sequences. Lithuanian Mathematical Journal, 56, 503-517.
  * [11] Gúzman, S. S., & Luca, F. (2014). Linear combinations of factorials and S-units in a binary recurrence sequence. Annales Mathématiques du Québec, 38(2), 169-188.
  * [12] Laurent, M., Mignotte, M., & Nesterenko, Y. (1995). Formes linéaires en deux logarithmes et déterminants d’interpolation. Journal of number theory, 55(2), 285-321.
  * [13] Lenstra, A. K., Lenstra, H. W., & Lovász, L. (1982). Factoring polynomials with rational coefficients. Mathematisches Annalen 261, 515–534.
  * [14] Matveev, E. M. (2000). An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II. Izvestiya: Mathematics, 64(6), 1217.
  * [15] Miles, E. P. (1960). Generalized Fibonacci numbers and associated matrices. The American Mathematical Monthly, 67(8), 745-752.
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  * [19] Smart, N. P. (1998). The algorithmic resolution of Diophantine equations: a computational cookbook (Vol. 41). Cambridge University Press.
  * [20] Wolfram, D. A. (1998). Solving generalized Fibonacci recurrences. Fibonacci Quarterly, 36(2), 129-145.

## Addresses

1 Department of Mathematics, School of Physical Sciences, College of Natural
Sciences, Makerere University, Kampala, Uganda

Email<EMAIL_ADDRESS>
Email<EMAIL_ADDRESS>
Email<EMAIL_ADDRESS>
2 School of Mathematics, Wits University, Johannesburg, South Africa and
Centro de Ciencias Matemáticas UNAM, Morelia, Mexico

Email<EMAIL_ADDRESS>
## Appendices

### Appendix 1

    memo = {}
    def k_generalized_lucas(n, k):
        if (n, k) in memo:
            return memo[(n, k)]
        # Base cases
        if n < 2-k:
            return 0
        elif n == 0:
            return 2
        elif n == 1:
            return 1
        else:
             # Recurrence relation
             result = sum([k_generalized_lucas(n-i, k) for i in range(1, k+1)])
             memo[(n, k)] = result
             return result
    def check_multiplicative_dependence(a, b):
        # Check for multiplicative dependence
        for x in range(1, n):
           for y in range(1, n):
                if x != y and a ^ x == b ^ y:
                     return True
        return False
    # Check for multiplicative dependence
    for k in range(2, 51):
       for n in range(2, 51):
           for m in range(2, n):  # Ensure m < n
               lnk = k_generalized_lucas(n, k)
               lmk = k_generalized_lucas(m, k)
               if check_multiplicative_dependence(lnk, lmk):
                  print(f"For k={k}, the numbers L_{n}^{(k)}={lnk} and L_{m}^{(k)}={lmk}
                  are multiplicatively dependent.")

### Appendix 2

    tau = log(3)/log(2)
    M = 10^114
    cf_tau = continued_fraction(tau)
    N = 0

    for convergent in cf_tau.convergents():
        if convergent.denominator() > M:
           break
        N += 1

    print(f"The index N such that q_N > M is: {N}")
    print(f"The convergent p_N/q_N is: {cf_tau.convergents()[N-1]}")

    a_M = max(cf_tau[:N+1])
    print(f"The value a(M) is: {a_M}")
    Ψ

### Appendix 3

    def k_generalized_lucas(n, k, memo={}):
        if (n, k) in memo:
           return memo[(n, k)]
        if n < 2 - k:
           return 0
        elif n == 0:
           return 2
        elif n == 1:
           return 1
        else:
           result = sum([k_generalized_lucas(n - i, k, memo) for i in range(1, k + 1)])
           memo[(n, k)] = result
           return result
    # Function to check the GCD conditions
    def check_gcd_conditions(n, m, k):
        ln = k_generalized_lucas(n, k)
        lm = k_generalized_lucas(m, k)
        gcd_val = gcd(ln, lm)
        if power_mod(gcd_val, n, ln) == 0 and power_mod(gcd_val, m, lm) == 0:
            return True
        return False
    # Function to check the multiplicative dependence
    def check_multiplicative_dependence(n, m, k):
        ln = k_generalized_lucas(n, k)
        lm = k_generalized_lucas(m, k)
        for x in range(1, n):
           for y in range(1, n):
               if x != y and ln ^ x == lm ^ y:
                   return True
        return False
    pairs = []
    for k in range(3, 1001):
        for n in range(4, 1600):
            for m in range(3, n):
                if check_gcd_conditions(n, m, k):
                      pairs.append((k, n, m))
    for k, n, m in pairs:
         if check_multiplicative_dependence(n, m, k):
              print(f"For k={k}, the pair (n={n}, m={m}) satisfies the
              multiplicative dependence condition.")

### Appendix 4

    from sage.all import *
    # Define alpha and C
    alpha = (1 + sqrt(5)) / 2
    C = 6e194  # 6 * 10^194
    Ψ
    # Define f_k(x) function
    def f_k(x, k):
        return (x - 1) / (2 + (k + 1) * (x - 2))
    # Initialize variables to store the best c1, delta, and corresponding k
    best_c1 = float(’inf’)  # Initialize to a large value
    best_delta = 0
    best_k = None
    Ψ
    # Loop over k values
    for k in range(3, 1001):  # Adjust the range as needed
        # Define matrix B for each k
        B = Matrix(QQ, [[1, 0, 0],
                       [0, 1, 0],
                       [floor(C * (log(f_k(alpha, k)).n())),
                        floor(C * (log(2*alpha - 1).n())),
                         floor(C * (log(alpha).n()))]])
    Ψ
        # Apply LLL reduction to get a reduced basis
        B_reduced = B.LLL()
    Ψ
        # Perform Gram-Schmidt orthogonalization on the reduced basis
        B_orthogonalized = B_reduced.gram_schmidt()[0]
    Ψ
        # Compute c1
        c1 = min([B_reduced.column(0).norm() / B_orthogonalized[i].norm()
        for i in range(B_reduced.ncols())])
    Ψ
        # Define y and the lattice L
        y = vector([0, 0, 0])  # Replace with your actual vector
        L = B_reduced * ZZ^B_reduced.ncols()
    Ψ
        # Compute lambda and delta
        if y not in L:
            z = B_reduced.inverse() * y
            i0 = max([i for i in range(len(z)) if z[i] != 0])
            lambda_val = abs(z[i0] - round(z[i0]))
        else:
            lambda_val = 1
    Ψ
        delta = lambda_val * B_reduced.column(0).norm() / c1
    Ψ
        # Update best_c1, best_delta, and best_k if a better c1 is found
        if c1 < best_c1:
           best_c1 = c1
           best_delta = delta
           best_k = k
    Ψ
    # Output the best c1, delta, and corresponding k
    print(f"Best k: {best_k}")
    print(f"Best c1: {float(best_c1):.6e}")
    print(f"Best delta: {float(best_delta):.6e}")


# Shaping Multi-Robot Patrol Performance with
Heterogeneity in Individual Learning Behavior ††thanks: CY is supported by UK
Foreign, Commonwealth & Development Office Services. ZRM is supported by a
University of Bristol PhD Scholarship. ERH is supported by the Royal Academy
of Engineering under the Research Fellowship program.

Connor York
School of Engineering
Mathematics & Technology
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>Zachary R. Madin
School of Engineering
Mathematics & Technology
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>Paul O’Dowd  School of Engineering
Mathematics & Technology
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>Edmund R. Hunt  School of Engineering
Mathematics & Technology
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>
###### Abstract

Individual differences in learning behavior within social groups, whether in
humans, other animals, or among robots, can have significant effects on
collective task performance. This is because it can affect individuals’
response to the environment and their interactions with each other. In recent
years there has been rising interest in the question of how individual
differences, whether in learning or other traits, affect collective outcomes:
studied, for example, in social insect foraging behavior. Multi-robot, ‘swarm’
systems have a heritage of bioinspiration from such examples, and here we
consider whether heterogeneity in a learning behavior called latent inhibition
(LI) may be useful for a team of patrolling robots tasked with environmental
monitoring and anomaly detection. Individuals with high LI can be seen as
better at learning to be inattentive to irrelevant or unrewarding stimuli,
while low LI individuals might be seen as ‘distractible’ and yet, more
positively, more exploratory. We introduce a simple model of the effects of LI
as the probability of re-searching a location for a reward (anomalous reading)
where it has previously been found to be unrewarding (irrelevant). In
simulated patrols, we find that a negatively skewed distribution of mostly
high LI robots, and just a single low LI robot, is collectively most effective
at monitoring dynamic environments. These results are an example of
‘functional heterogeneity’ in ‘swarm engineering’ and could inform predictions
for ecological distributions of learning traits within social groups.

###### Index Terms:

latent inhibition, learning, multi-robot patrol, collective behavior, anomaly
detection

##  I Introduction

Collective behavior is everywhere in the human and animal world, because
cooperation allows social groups to achieve goals that would be difficult or
impossible for an individual [43]. For robotics engineers, there are many
challenging environments where task complexity or time constraints favor the
deployment of cooperative teams of robots [33]. A distinctive approach to
multi-robot systems design is swarm robotics, which takes inspiration from
biological self-organization, especially that of social insect societies, to
implement decentralized cooperative behaviors [36, 40]. Mathematical and
simulation models of large-scale collective behavior have tended to assume
homogeneity in agent traits [4, 43, 15]. While this is a logical starting
point for scientific investigation and bioinspired engineering, heterogeneity
within social groups such as the social insect ‘superorganism’ is increasingly
recognized as having important functional benefits [32]. An excellent example
of such heterogeneity is observed in honeybee colony (Apis mellifera)
homeostasis, where genetic diversity favors enhanced nest thermoregulation via
variation in temperature response thresholds [21, 31]. Another likely example
of adaptive individual variation is in cognition: honeybees exhibit variation
in a learning behavior known as latent inhibition (LI) [8]. LI is a form of
learning to ignore unimportant information – in the case of honeybees, this
could be unrewarding odors [12]. It has been studied in vertebrates with
respect to predator recognition [28, 13], for instance, and in humans low LI
is associated with attention disorders [38]. A recent study of honeybees by
Cook et al. exploited the genetic heritability of LI to create artificially
manipulated colonies with only high-LI, only low-LI and 50/50 mixed high- and
low-LI worker bees. The colonies’ collective foraging behavior was then
compared to non-selected control colonies [7]. The study found that high-LI
colonies preferred to visit familiar food locations, whereas low-LI colonies
visited familiar and unfamiliar locations equally. In mixed-LI colonies, low-
LI individuals were influenced by high-LI individuals, via their intensive
recruitment efforts, to visit familiar locations. Thus, differences in
individual cognition – attention to important information, i.e. known food
locations – and inter-individual interactions were both found to be
significant in driving overall collective behavior, or what might be called
‘collective cognition’ [9]. The distribution of LI within natural colonies,
and its influence on novelty seeking [25], is hypothesized to help manage
exploration–exploitation trade-offs over time and space in dynamic
environments [7]. Thus, the variation underlying this distribution is
presumably under natural selection [2], having important adaptive
significance.

Here, we are inspired by the opportunity for multi-robot systems engineering
to introduce variation in individual robot learning behavior, with the
strategic aim of obtaining collective learning phenotypes that better manage
exploration–exploitation trade-offs. We also consider that such multi-robot
systems research could have relevance to understanding whether and how human
and animal collectives take advantage of functional heterogeneity [32]. The
application domain of our experiments is persistent environmental monitoring
for security purposes – known as robotic patrolling [1]. Regular patrolling
requires robots to visit important locations as frequently as possible, while
adversarial patrolling seeks to introduce an element of unpredictability in
movements to make it harder for an attacker to circumvent the system [17]. Our
starting point here is regular patrolling, where one or more robots have to
routinely inspect certain sensitive parts of the environment, in return for a
‘reward’ reflecting user preferences on performance. Beyond the typical
patrolling problem, we consider that each location has a number of targets
that can be selectively inspected for a reward – thus the task somewhat
resembles the well-known exploration–exploitation problem [24] in that robots
must choose to either inspect known rewarding targets or invest more time in
re-inspecting potentially uninformative targets. Locations with clusters of
targets could be seen as analogous to co-located ‘foraging patches’ where
honeybees might collect food (nectar, pollen). Our main hypothesis is that
multi-robot systems that comprise mainly ‘high-LI’ individuals alongside a few
‘low-LI’ individuals, will best embody an effective patrol strategy – if low-
LI individuals can share newly-acquired information. We carry out both agent-
based simulations and real robot trials, for various group sizes and
individual LI compositions; this is described in section III, Methods, before
which we provide more background on LI and robot environmental monitoring in
section II. Results are presented in section IV and discussed in section V.

##  II Background

###  II-A Latent Inhibition and Attention to Relevant Information

Although latent inhibition (LI) as a phenomenon has been extensively studied
over the past 60 years or so, there is not yet a consensus on its
mechanism(s), with a variety of theoretical models proposed [16]. The typical
demonstration of an LI effect involves tracking the acquisition of conditioned
response (CR) to a conditioned stimulus (CS) that is followed by an
unconditioned stimulus (US). LI is observed as a delay or decrement to the
appearance of the CR to a CS that had previously been presented on its own
[16]. This indicates that something has been learned during initial exposure
to the CS that interferes with subsequent learning about the CS, or with the
expression of that learning [26]. The most popular explanation of LI is that
the pre-exposure to the CS, without any reinforcement, results in a continuing
reduction in attention to it; although there could be multiple processes at
work to produce LI’s effects [16]. The mechanisms underlying an individual’s
cognitive processes of information processing and decision-making are
increasingly recognized as having important ecological significance [10], and
this is presumably true of biological and robotic collectives as well,
especially when one considers the broad scope for realizations of inter-
individual variability. The potential of neural network models of adaptive
behavior, for example, to be implemented in robots and exhibit natural
phenomena such as LI has been noted for some time [39], and such effects have
been demonstrated in Nao humanoid robots [3]. For the purpose of our robot
model of LI, it seems unnecessary to commit to, and implement, a certain
theoretical model of LI – for our purposes, we are primarily inspired by the
potentially useful effects of variation in LI. Here the effect of interest,
following findings such as Cook et al. [7], is variable attention to important
(mission-relevant) information; this has also been described in terms of
‘behavioral persistence’ and ‘site fidelity’ in the context of collective
foraging [30]. In the case of robot patrolling, this could relate to the
location of important points of interest, or ‘hotspots’, that ought to be
checked regularly. Thus, we implement a simple model of LI, which reflects
attention to immediate rewards, as represented by a probability of re-scanning
an ‘irrelevant’ (unrewarding) target. This is somewhat akin to high-LI
honeybees focusing on familiar food locations that are known to be rewarding.
The LI model is described more in the Methods.

###  II-B Environmental Monitoring and Multi-Robot Patrol

Robot systems are a popular tool for environmental monitoring across wide
spatial and temporal scales, for purposes such as scientific research,
pollution tracking, or ecological survey [11]. Robots are also increasingly
used for long-term security monitoring, with the central framing of the
problem being multi-robot patrol (MRP), where robots must regularly pass
through designated locations of interest [1]. The paths between locations are
commonly described as a graph, which can be input into a patrolling algorithm
that produces target destinations for robots [34]. The focus of MRP algorithms
is on the effective coordination of robots such that the nodes on the patrol
graph are visited as often as possible; and/or on intelligent individual
decisions about where to visit next [34]. While such algorithms can be
deterministic in their outcomes, machine learning methods such as
reinforcement learning have also be proposed (e.g. [37]) which allow for
online adaptation to the environment. The focus of the MRP problem on
‘idleness’ minimization (the time between visits to a node) tends to overlook
some practical concerns a user might have in a real-world security
application. First of all, in an adversarial context, deterministic robot
behaviors will be easy to subvert, and so introducing an element of
stochasticity to decision-making will be helpful [17] (one might also wish to
benchmark MRP algorithm performance against an attacker model [44]). In this
regard, introducing heterogeneity in robots’ behavioral processes could be
seen as inherently worthwhile to contribute to system unpredictability. A
second practical concern is that locations on the map may require more time
investment than a momentary visit. For example, when a robot physically
arrives at a location, there may be a choice of how thorough a search to
conduct, such as an inspection of one or more entities present at that
location, which requires focused attention on that entity for a period of
time. For instance, imagine a local search of the corner of an office
building: one might have a choice of inspecting a computer monitor, a network
switch, a wastepaper basket, or a photocopier. Upon first inspection of the
wastepaper basket, there may be nothing ‘rewarding’ present (no anomalous
sensor readings), whereas the computer monitor may produce a signal that is
distinctive enough to merit a repeat inspection at a later time. This aspect
of search thoroughness has not, to our knowledge, been represented in the MRP
problem; it could be seen as a particular example of the ubiquitous
‘exploration–exploitation’ problem [24, 27], although in this case it is not
that targets require ‘exploration’ (their presence is known) but rather, re-
inspection. This problem can be framed as ’multi-armed bandit’ problem (e.g.
[23]), which when rewards change over time can be referred to as a ’restless
bandit’ [42]. The need to apply selective attention to relevant targets is
thus where we perceive the latent inhibition concept to be potentially useful
for MRP. When targets are found to be worthy of inspection, one would wish for
robots to re-inspect them at a later time, and if they are not, one would wish
for robots not to waste time coming back to them frequently at the expense of
idleness minimization. However, in a dynamic environment, some level of re-
inspection is necessary, to ensure the system’s performance does not suffer
from stale information [24]. A strategy for managing this problem would be to
introduce a distribution of selective attention ability, such that most robots
preferentially re-inspect rewarding targets. Such group-wide variation could
be fixed before deployment, given knowledge about generally effective
distributions, or adapt online with robots’ experiences. We explain our
approach further in the next section.

##  III Methods

We carry out experiments with an agent-based patrolling simulator up to $N=6$
robots in an office environment. We describe the implementation of a model of
‘latent inhibition’ in the patrolling robots, with respect to their mission of
regularly visiting locations of interest and inspecting appropriate targets.
We then describe the simulator and the experimental trials.

Figure 1: Map and patrol route (dark green) used during the simulations. Stars
represent the robots’ starting positions, and circles represent the nodes.
Colored lines show the robots’ paths onto the patrol route for $N=6$. The map
was obtained from 2D lidar mapping of our own office environment.

###  III-A Latent inhibition model and patrol strategy

Robots travel between user-designated points of interest in the environment
(waypoints in an ongoing patrol). At each waypoint there are four targets of
potential interest that the robot can inspect. The existence of the targets is
known but to inspect each one takes an investment of time, representing, for
example, a scanning process with the robot’s sensors. Here we set a fixed scan
time corresponding to around $S_{T}=8$ s in simulation. Thus, robots have to
choose between scanning targets at a certain location or traveling on promptly
to the next location. If a target has never been scanned before by a robot, as
is the case for all targets at the beginning of an experimental trial, it will
scan it to ascertain whether it is noteworthy (e.g., potential to produce
anomalous readings, or vulnerability to attack). To represent this process in
the experiments, we set half of the targets to produce a reward ($+1$ reward)
when scanned. Robots maintain a memory of whether a target has produced a
reward, and if it has produced a reward on the last time of scanning, it will
be re-scanned the next time that robot visits the waypoint. If it did not
produce a reward, there is a probability $p_{r}$ of it being re-scanned, which
is simply:

$p_{r}=1-\text{LI}$ (1)

where $\text{LI}\in[0,1]$. That is, low LI individuals have a high probability
of re-scanning previously unrewarding targets, and high LI individuals have a
low probability of re-scanning them. This represents the narrowly focused
attention to immediately relevant information observed in high LI individuals,
as seen, for example, in high-LI honeybees’ attention to known food sources,
or conversely the broader attention to possible rewards of low LI honeybees
[7]. The robots refer to a lookup table of actual rewards, representing the
ground truth, which is either fixed for the duration of a trial or may change
one or more times, reflecting a dynamic environment. When robots scan a target
they immediately and accurately update their belief of whether it is
rewarding.

The robot patrolling algorithm used here is simply a cyclic strategy, i.e. the
robot(s) follow a path that visits all nodes on the patrol graph [34]. Because
we begin with a small patrol graph, all robots follow the same path, unlike a
partitioning strategy which would divide the graph up into subgraphs for each
robot, which is better suited to maps with separated regions [34]. Robots
commence a trial physically separated across the map (Fig. 1), but because all
of the robots follow the same path, there is scope for robots to catch up and
even overtake each other, with consequent scope to interfere with each other’s
movements.

###  III-B Agent-based robot simulations

The Python simulation model is implemented as a two-dimensional grid-world
(Fig.1), generated from a real 2D lidar map with each cell representing
approximately 2 cm in the real world. On the map locations of interest are
represented as nodes within a connected graph. The graph is defined by the
adjacency matrix, which is an $W\times W$ matrix where $W$ is the number of
nodes in the graph. The weightings between nodes are the lengths of the paths
between nodes in the grid-world. In a single time step, robots are able to
move in any compass heading from the current cell that they occupy, including
diagonal transitions to the eight surrounding cells. Diagonal cell transitions
have a $\sqrt{2}$ transition cost, or $1$ otherwise. Each robot occupies a
single cell at a time, and executes a patrolling algorithm (here, simply a
cyclic patrol route) to determine which node in the graph to visit next,
according to the adjacency matrix of the graph. Once an agent has determined
which node to visit, a shortest path implementation of A* path planning is
used to reach the destination. The simulation was calibrated to produce
similar results to an experimental proof-of-concept with real robots (‘Leo
Rovers’) in the same environment: initial trials indicated that 30 minutes of
real patrolling corresponded to around 1300 simulation steps. Robots move at
20 cells per step, or around $0.4$ m/s. Physical inter-robot interference is
represented by an occupancy rule that only one robot is allowed in a cell at
one time. If a cell that a robot is trying enter is occupied (any cell, not
just a waypoint), then the robot will delay for 3 simulation steps, roughly
equivalent to 4 seconds, before retrying. As the kinematics of the agents are
not modeled, nor the sensor or localization models, the simulator allows fast
investigation of multiple system configurations.

We modeled communication between robots by enabling a shared set of beliefs on
the reward of targets, which updates immediately when a reward is received.
Thus, rather than each robot having a private set of beliefs on the location
of rewarding targets, based on its own individual experience, when
communication is enabled as soon as a target is found to be rewarding all
robots in the group share that belief. This means that a low LI robot can have
a significant impact on the group’s beliefs, as re-scanning a previously
unrewarding target and finding it to be rewarding can not only affect its own
future choices about where to scan, but the whole group’s choices.

###  III-C Experimental trials

Figure 2: Average system reward for $N\in\\{1,2,4,6\\}$ robots by group LI
composition. White is communication disabled, blue is communication enabled.
Top row: static reward environment. Middle row: 1 environment shift. Bottom
row: 3 environment shifts.

The experiment was set up as a cyclic patrol route followed between a certain
number of waypoints, $W$, with 4 targets placed at each location. Therefore,
there are $4W$ targets available for each robot to scan on each cycle of the
patrol route. Of these $4W$ targets, we set half as providing a $+1$ reward,
with the other half providing $0$ reward. This assignment is made randomly but
is fixed across trials. We used a map of a real-world office environment (a
university office space), and set 4 waypoints, so there were a total of 16
targets to be scanned (Fig. 1). To test the effects of a changing environment
on the system, in some experimental treatments, after a certain period of time
($T/3$ for 1 switch, or equally spaced $T/(S+1)$ periods for $S$ switches),
the rewards assigned to the targets are changed. 50% switch their reward (from
rewarding to unrewarding or vice versa) and 50% of the tags maintain their
previous reward. In simulation trials range from 0 switches (static
environment) up to 3 switches (highly dynamic environment). We anticipate that
more dynamic environments will favor robot teams comprising one or more low LI
individuals, that will be more likely to re-scan previously unrewarding
targets.

For the trials we used $N\in\\{1,2,4,6\\}$. For variation in our latent
inhibition model, we investigated both homogeneous and heterogeneous group
compositions of LI, using ‘low’ (L) and ‘high’ (H) LI as
$\text{LI}\in\\{0.5,0.95\\}$. This corresponds to a probability of re-scanning
previously unrewarding targets, $p_{r}$, of $0.5$ and $0.05$ respectively. For
heterogeneous compositions we examined either 50/50 mixtures of both types, or
a single low or high LI individual among a group of the other type. For
example, for $N=4$ we simulated groups of $\\{0.5,0.5,0.5,0.95\\}$,
$\\{0.5,0.5,0.95,0.95\\}$, and $\\{0.5,0.95,0.95,0.95\\}$. This was to examine
the possible non-linear impact of a single heterogeneous individual on group
collective performance. We repeated trials both with and without communication
between robots, carrying out 10 simulated trials per configuration, for 1300
simulated time steps. We examine the total reward gathered by the robot system
after that time to assess system performance. We use Tukey’s HSD test for
pairwise mean performance comparison across system LI compositions.

##  IV Results

Figure 3: Average system reward over time for 6 robots. Top: Static reward
environment, communication enabled. Middle: 3 environment shifts,
communication disabled. Bottom: 3 environment shifts, communication enabled.

In a static environment, where the reward status of targets is unchanged for
the duration of a trial, individuals or groups with high LI (low attention to
previously unrewarding targets) obtain the most reward. This is because there
is no benefit to low LI, which only serves to slow robots down in their patrol
between waypoints because of frequent, unnecessary re-scanning. There is a
small benefit to communication, because early in a patrol robots can share the
location of unrewarding targets, thus saving some re-scan time for high LI
individuals, allowing them to make faster initial progress around the patrol
route (Fig. 2, top row, Fig. 3, top).

When there is a one-off shift in the reward locations (50% of targets
changing: 25% becoming rewarding and 25% becoming unrewarding), there can be a
benefit to having a lower LI (Fig. 2, middle row). Starting with the no
communication case, for $N=2$ robots, for example, the 2L and 1L1H combination
have a higher average total reward than 2H (Tukey post hoc:
$Z=5.1,P=0.035;Z=7.6,P=0.001$) though 2L and 1L1H are not different to each
other ($Z=-2.5,P=0.41$). Generally though, the advantage of lower LI is not
dramatic, because the benefit of successfully identifying newly-rewarding
targets is balanced against the slower progress around the patrol route. When
communication is enabled, however, the total reward is higher across the board
for 1 environment switch, because robots have the benefit of sharing any new
information they obtain. Re-examining $N=2$, 1L1H with communication enabled
performs significantly better than 2L ($Z=14,P<0.001$), because it has the
advantage of one higher LI individual to promptly continue a patrol route
between known rewards, including those identified by the lower LI individual;
it also outperforms 2H ($Z=7.2,P=0.005$). Examining the larger group sizes,
for $N=4$, the compositions with 4H, 1L3H, 2L2H perform better than 4L, 3L1H,
although they are not significantly different to each other. For example, 1L3H
does not perform better than 2L2H ($Z=9.5,P=0.063$) or 4H ($Z=5.4,P=0.53$).
Nevertheless, although 1L3H and 4H have similar means, 4H can be seen as
having larger variation in performance (Fig. 2, $N=4$, middle row). This is
because with 4 high LI robots, there is the possibility, though not certainty,
for one or more robots to successfully re-scan relevant (changed) targets and
share that information with the others. For $N=6$, the group composition with
a single low-LI individual (1L5H) is significantly more successful than the
50/50 composition ($Z=26.5,P<0.01$), unlike the $N=4$ team trials. This is
presumably because the larger team has the benefit of one low LI individual
re-scanning without too many low LI individuals to slow down its overall
patrolling between waypoints and known rewarding targets. Even so, for $N=6$,
with only a one-off shift in target rewards, it remains the case that 1L5H and
6H are not significantly different in performance ($Z=5.9,P=0.687$) – the
change is not sufficiently disruptive to the robots’ knowledge of the
environment that six high-LI individuals cannot maintain comparable
performance.

When there is a highly dynamic environment, with three successive shifts in
target rewards, the benefit of a negatively skewed distribution of LI is clear
for larger group sizes, when communication is enabled (Fig. 2, bottom row).
For $N=6$, 1L5H clearly outperforms 6H ($Z=44.9,P<0.01$) and also the 50/50
composition, 3L3H ($Z=12.4,P=0.047$). Without communication, there is a linear
increase in performance the more low LI individuals are present in the group –
because robots are effectively working alone, they need to have low LI to
notice the shift in the environment reward distribution (Fig. 3: compare
middle and bottom).

##  V Discussion

We simulated systems of patrolling robots with communication alternatively
enabled or disabled. At each waypoint on the patrol route, there were 4
targets to be scanned, and robots varied in their attention to scanning
previously unrewarding targets. This was inspired by the phenomenon of latent
inhibition (LI) in humans and animals [16, 28], whereby individuals with high
LI tend to be better at focusing their attention on relevant information [7]:
in this case, not re-inspecting previously unrewarding targets. Nevertheless,
in dynamic environments, sometimes it can be adaptive to reexamine such
targets, because their status may have changed with the passage of time. In a
social insect system (honeybee colonies), the distribution of LI within
natural colonies (and its influence over novelty seeking) is hypothesized by
Cook et al. to help manage exploration–exploitation trade-offs over time and
space in dynamic environments [7]. In the context of multi-robot systems or
swarm engineering, the impact of variation in such learning behavior could be
highly relevant for shaping system strategy in approaching tasks related to
such trade-offs. Our main hypothesis was that systems comprising mostly, but
not exclusively, high LI individuals, will be generally most effective at
monitoring dynamic environments – if they can communicate freely. This is
because the low LI individuals could share information gained from more
exploratory behavior, while high LI individuals could focus their attention on
known rewarding targets. In natural honeybee colonies, collective behavior is
found to resemble that of artificially composed low-LI colonies [7], which is
coherent with natural selection for a highly dynamic environment. Here, we
found that simulated robot systems with compositions of low LI individuals
also performed best in highly dynamic environments. When communication between
individuals is disabled, there is a clear correspondence between number of low
(high) LI individuals and performance in dynamic (static) environments,
because individuals are having to rely entirely on private information to make
their target choices. In larger group sizes, when communication is enabled,
the addition of only one low-LI individual to a group of high-LI robots led to
a significant performance increase: an example of the nonlinear impact of
heterogeneous individual behavior on collective behavior, an effect which is
increasingly recognized as ubiquitous in the natural world [35, 29, 5].

A recent review of the mechanisms of collective learning by Collet et al.
identified three distinct ways through which groups can improve their
collective task performance [6]. These are: individuals learning to better
solve tasks alone; members learning about each other’s behavior to be better
coordinated; and learning to better complement others through specialization
of their own behavior [6]. To obtain the benefits of a suitable LI
distribution observed in this study, according to the experienced extent of
environmental change, robots could be endowed with some level of plasticity in
their LI value. This would be the third form of learning identified by Collet
et al. (complementarity via specialization). Because LI is commonly regarded
as a stable cognitive trait, adaptability in LI could be regarded as some form
of phenotypic plasticity, which we have previously considered in respect of
‘personality’ variation [20, 19, 18], another aspect of cognitive variation
[14, 41]. Specialization implemented through models such as response threshold
reinforcement can be highly effective at improving task performance, yet in
dynamic environments flexibility needs to be retained somehow to allow
respecialization [22]. In future work we will consider individual flexibility
in LI/attention traits for obtaining adaptive group-level distributions and
thus achieving effective collective learning in the sort of task we have
examined here, such as environmental monitoring. The link between models of
individuals’ cognitive processes and physical action in collective behavior
research is relatively weak, arguably in both biology and robotics [15, 45],
with ample scope to better reflect the diversity and dynamism of individual
differences in traits such as learning and personality. We plan to examine
this further, including with real robot experiments, which may help to shed
light on commonalities in adaptive embodied collective cognition in humans,
animals and robots.

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