Florian Luca

• PhD
• Professor at University of the Witwatersrand

Let $$ \{L_n\}_{n\ge 0} $$ { L n } n ≥ 0 be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $$L_n-2^x3^y=c$$ L n - 2 x 3 y = c , for $$n,x,y\in \mathbb {Z}_{\ge 0}$$ n , x , y ∈ Z ≥ 0 . We treat the cases $$c\in -\mathbb {N}$$ c ∈ - N , $$c=0$$ c = 0 and $$c\in \mathbb {N}$$ c ∈ N independently. In the...

Let $k\ge 2$ and $\{F_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$--generalized Fibonacci numbers whose first $k$ terms are $0,\ldots,0,0,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we determine all $k$-Fibonacci numbers that are palindromic concatenations of two distinct repdigits.

In this paper, we find all solutions of the Diophantine equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n^x+F_k^x=F_m^y$$\end{document}, where \documentclass...

Let $k\ge 2$ and $\{L_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$-generalized Lucas numbers whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we show that this sequence does not contain the discriminant of its characteristic polynomial.

We introduce the notion of a twisted rational zero of a nondegenerate linear recurrence sequence (LRS). We show that any nondegenerate LRS has only finitely many such twisted rational zeros. In the particular case of the Tribonacci sequence, we show that 1/3$1/3$ and −5/3$-5/3$ are the only twisted rational zeros that are not integral zeros.

We prove transcendence of the Hecke–Mahler series , where is a non‐constant polynomial, is a real number, is an irrational real number and is an algebraic number such that .

In this paper, we study the Diophantine equation $$ b^k + \left(a+b\right)^k + \left(2a+b\right)^k + \cdots + \left(a\left(x-1\right) + b\right)^k = y\left(y+c\right) \left(y+2c\right) \cdots \left(y+ \left(\ell-1\right)c\right),$$ where $a,b,c,k,\ell$ are given integers under natural conditions.~We prove some effective results for special values f...

We prove transcendence of the Hecke-Mahler series $\sum_{n=0}^\infty f(\lfloor n\theta+\alpha \rfloor) \beta^{-n}$, where $f(x) \in \mathbb{Z}[x]$ is a non-constant polynomial $\alpha$ is a real number, $\theta$ is an irrational real number, and $\beta$ is an algebraic number such that $|\beta|>1$.

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation $\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\lef...

Let {Tn}n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{T_n\}_{n\ge 0} $$\end{document} be the sequence of Tribonacci numbers. In this paper, we study the exponen...

We prove that for any integers $\alpha, \beta > 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ is decidable (where $\alpha^{\mathbb{N}}$ is the set of positive integer powers of $\alpha$, and likewise for $\beta^{\mathbb{N}}$). On the other hand,...

Let ( P m ) m ≥0 be the sequence of Pell numbers given by P 0 = 0, P 1 = 1, and P m +2 = 2 P m +1 + P m for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x ² − dy ² = ± 1, which is a product of two Pell numbers.

The integer sequence defined by P_{n+3}=P_{n+1}+P_{n} with initial values P_{0}= P_{1}=P_{2}=1 is known as the Padovan sequence (P_{n})_{n∈Z}. In this note, we solve the Diophantine equations P_{−n}=±P_{m}^{2}, P_{n}=P_{−m}^{2}, and P_{−n} =±P_{−m}^{2} in positive integers n, m.

The k –generalized Fibonacci sequence $$\{F_n^{(k)}\}_{n\ge 2-k}$$ { F n ( k ) } n ≥ 2 - k is the linear recurrent sequence of order k whose first k terms are $$0, \ldots , 0, 1$$ 0 , … , 0 , 1 and each term afterwards is the sum of the preceding k terms. The case $$k=2$$ k = 2 corresponds to the well known Fibonacci sequence $$\{F_n\}_{n\ge 0}$$ {...

Let $$(L_n^{(k)})_{n\ge 2-k}$$ ( L n ( k ) ) n ≥ 2 - k be the sequence of k –generalized Lucas numbers for some fixed integer $$k\ge 2$$ k ≥ 2 whose first k terms are $$0,\ldots ,0,2,1$$ 0 , … , 0 , 2 , 1 and each term afterward is the sum of the preceding k terms. For an integer m , let P ( m ) denote the largest prime factor of m , with $$P(0)=P(...

In 1953, Carlitz showed that all permutation polynomials over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb F}_q$$\end{document}, where \documentclass[12pt]{...

Let {Un}n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{U_n\}_{n\ge 0} $$\end{document} be the Lucas sequence. For integers x, n and m, we find all solutions to U...

The well-known Fibonacci sequence has several generalizations, among them, the k-generalized Fibonacci sequence denoted by F (k) . The first k terms of this generalization are 0, . . . , 0, 1 and each one afterward corresponds to the sum of the preceding k terms. For the Fibonacci sequence the formula F_{n+1}^{2}-F_{n-1}^2=F_{2n} holds for every n ≥...

Let S be a finite, fixed set of primes. In this paper, we show that the set of integers c which have at least two representations as a difference between a factorial and an S-unit is finite and effectively computable. In particular, we find all integers that can be written in at least two ways as a difference of a factorial and an S-unit associated...

In this paper we consider a (non)congruence generalizing the so-called good/bad numbers introduced by Moree (Acta Arith LXXX 3:197–212, 1997) and give asymptotics for their counting functions. In addition, we give heuristics for some conjectured bounds on primes belonging to such a class.

We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ for $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ a Sturmian sequence over a binary alphabet and $\beta$ an algebraic number with $|\beta|>1$. We show that every such number is transcendental. More generally, for a given base~$\beta$ and given irrat...

We revisit the classical subject of equidistribution of the roots of Littlewood-type polynomials. More precisely, we show that the roots of the family of polynomials Ψ k ( z ) = z k −z k− 1 −⋯−1, k ⩾ 1, are uniformly distributed around the unit circle in the strong quantitative form, confirming a conjecture from [C.-A. Gómez and F. Luca, Commentat....

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, (weighted) automata theory, and the analysis of linear dynamical systems, amongst many others. Decidability of the Skolem Problem is notoriously o...

We find all the solutions of the Diophantine equation Pn±a(10m-1)9=k!\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_n\pm \frac{a(10^m-1)}{9}=k!$$\end{document}, in p...

There are many results in the literature concerning linear combinations of factorials among terms of linear recurrence sequences. Recently, Grossman and Luca provided effective bounds for such terms of binary recurrence sequences. In this paper we show that under certain conditions, even the greatest prime divisor of un-a1m1!-⋯-akmk!\documentclass[...

Here, we find all positive integer solutions of the Diophantine equations in the title, where $(P_n)_n≥0$ is the Pell sequence $P_0=0$, $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $n≥0$. .

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective r...

Let \( \{F_n\}_{n\geq 0} \) be the sequence of Fibonacci numbers and let \(p\) be a prime. For an integer \(c\) we write \(m_{F,p}(c)\) for the number of distinct representations of \(c\) as \(F_k-p^\ell\) with \(k\ge 2\) and \(\ell\ge 0\). We prove that \(m_{F,p}(c)\le 4\).

Fn+13+Fn3-Fn-13=F3n, for all n≥1, is a well-known identity fulfilled by the Fibonacci numbers. In this paper, we study a variant of this identity, replacing the cubic powers by a general power on the left–hand side and allowing an additional unknown exponent on the right–hand side.

Let ( T n ) n ∈ Z (T_n)_{n\in {\mathbb Z}} be the Tribonacci sequence and for a prime p p and an integer m m let ν p ( m ) \nu _p(m) be the exponent of p p in the factorization of m m . For p = 2 p=2 Marques and Lengyel found some formulas relating ν p ( T n ) \nu _p(T_n) with ν p ( f ( n ) ) \nu _p(f(n)) where f ( n ) f(n) is some linear function...

We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values

The integer sequence defined by P_{n+3}=P_{n+1}+P_{n} with initial conditions P_{0}=1 and P_{1}=P_{2}=0 is known as the Padovan sequence {P_n}_{n∈Z}. A recurrence sequence {u_n}_{n∈Z} is said to be of Padovan-type if it satisfies the same recurrence relation as the Padovan sequence but with arbitrary initial values u_{0}, u_{1}, u_{2} not all zero....

In this paper, we improve a result of Fujita and Le concerning the Diophantine equation x2+(2c-1)y=cZ

In this paper, we prove the theorem announced in the title.

Model checking infinite-state systems is one of the central challenges in automated verification. In this survey we focus on an important and fundamental subclass of infinite-state systems, namely discrete linear dynamical systems. While such systems are ubiquitous in mathematics, physics, engineering, etc., in the present context our motivation st...

Let d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document} be an integer which is not a square. We show that if (Ln)n≥0\documentclass[12pt]{minimal}...

Let $ \{F_n\}_{n\ge 0} $ be the sequence of Fibonacci numbers and let $p$ be a prime. For an integer $c$ we write $m_{F,p}(c)$ for the number of distinct representations of $c$ as $F_k-p^\ell$ with $k\ge 2$ and $\ell\ge 0$. We prove that $m_{F,p}(c)\le 4$.

The well-known Fibonacci sequence has several generalizations, among them, the k-generalized Fibonacci sequence denoted by F(k). The first k terms of this generalization are 0, ..., 0, 1 and each one afterward corresponds to the sum of the preceding k terms. For the Fibonacci sequence the formula F_{n+1}^2 - F_{n-1}^2 = F_{2n} holds for every n>0....

Here, we show that if u_n = n2n±1, then the largest prime factor of u_n±m! for n ≥ 0, m ≥ 2 tends to infinity with max{m,n}. In particular, the largest n participating in the equation un±m!=2^a3^b5^c7^d with n≥1, m≥2 is n=8 for which (8·2^8+1)−4!=34·52.

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (m, n, x, y), where for a positive integer k, Pk is the kth term of the Pell sequence.

It is conjectured that the sum Sr(n)=∑k=1nkk+rnk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_r(n)=\sum _{k=1}^{n} \frac{k}{k+r}\left( {\begin{array...

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech The...

The k-generalized Fibonacci and Pell polynomials are the polynomials Xk − Xk−1 − Xk−2 −⋯ − 1 and Xk − 2Xk−1 − Xk−2 −⋯ − 1, respectively. Here, k ≥ 2 is any integer. In this paper, we show that any two roots of some generalized Fibonacci and Pell polynomials are multiplicatively independent confirming a conjecture from [Bravo, Herrera and Luca, Comm...

Let $b$ be an algebraic number with $|b|>1$ and $\mathcal{H}$ a finite set of algebraic numbers. We study the transcendence of numbers of the form $\sum_{n=0}^\infty \frac{a_n}{b^n}$ where $a_n \in \mathcal{H}$ for all $n\in\mathbb{N}$. We assume that the sequence $(a_n)_{n=0}^\infty$ is generated by coding the orbit of a point under an irrational...

In this note we show that if $(u_n)_{n\geqslant 1}$ is a simple linearly recurrent sequence of integers whose minimal recurrence of order $k$ involves only positive coefficients that has positive initial terms, then $(Mu_{n^s})_{n\geqslant 1}$ is the sequence of periodic point counts for some map for a suitable positive integer $M$ and $s$ any suff...

We show that the $Kn$ –smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers n .

We show that the $Kn$--smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers $n$.

In this note, we look at the Diophantine equation obtained imposing that a Fibonacci number is a signed sum of values of the Ramanujan τ-function in factorials.

Let \((F_n)_{n\geqslant 0}\) and \((P_n)_{n\geqslant 0}\) be the Fibonacci and the Padovan sequences given by the initial conditions \(F_0=0\), \(F_1=1\), \(P_0=0\), \(P_1=P_2=1\) and the recurrence formulas \(F_{n+2}=F_{n+1}+F_n\), \(P_{n+3}=P_{n+1}+P_n\) for all \(n\geqslant 0\), respectively. In this note we study and completely solve the Diopha...

In this note, we that if { Fn(k) }n≥0 denotes the k-generalized Fibonacci sequence then for n ≥ 2 the closest integer to the reciprocal of ∑m≥n1/Fm(k) is Fn(k)−Fn−1(k) .

Let (Fn)n≥0 be the Fibonacci sequence given by F0=0,F1=1 and Fn+2=Fn+1+Fn,foralln≥0. In this paper, we find all positive integer solutions (m,n,a,k) of the Diophantine equation Fn±a(10m−1)9=k! with 1≤a≤9. Our proof requires lower bounds for nonzero linear forms in two logarithms of algebraic numbers both in the complex and p-adic cases and some com...

We extend an inequality for Fibonacci numbers published by P. G. Popescu and J. L. Díaz-Barrero in 2006.

Here we look at the Diophantine equation obtained by imposing that the Euler function of a member of the Y-coordinates of the Pell equation X2-dY2=±1,±4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidem...

We investigate the Diophantine equation x^2 −kxy + ky^2 + ly = 0 for integers k and l with k even. We give a characterization of the positive solutions of this equation in terms of k and l. We also consider the same equation for other values of k and l.

Let $E$ be an elliptic curve over the finite field $\mathbb F_q$. We prove that, when $n$ is a sufficiently large positive integer, $\#E(\mathbb F_{q^n})$ has a prime factor exceeding $n\exp(c\log n/\log\log n)$.

Let \(r\ge 1\) be an integer and \(\mathbf{U}:=(U_{n})_{n\ge 0} \) be the Lucas sequence given by \(U_0=0\), \(U_1=1, \) and \(U_{n+2}=rU_{n+1}+U_n\), for all \( n\ge 0 \). In this paper, we show that there are no positive integers \(r\ge 3,~x\ne 2,~n\ge 1\) such that \(U_n^x+U_{n+1}^x\) is a member of \(\mathbf{U}\).

In this paper, we survey some results concerning the occurrence of interesting arithmetic numbers, like Fibonacci numbers and repdigits in X-coordinates of Pell equations.

Let r≥1 be an integer and U≔{Un}n≥0 be the Lucas sequence given by U0=0,U1=1, and Un+2=rUn+1+Un for n≥0. In this paper, we explain how to find all the solutions of the Diophantine equation, AUn+BUm=CUn1+DUm1, in integers r≥1, 0≤m<n,0≤m1<n1, AUn≠CUn1, where A,B,C,D are given integers with A≠0,B≠0, m,n,m1,n1 are nonnegative integer unknowns and r is...

In this study, we find all Fibonacci and Lucas numbers which can be written as a difference of two repdigits. It is shown that the largest Fibonacci and Lucas numbers which can be written as a difference of two repdigits are $$\begin{aligned} F_{11}=89=111-22 \end{aligned}$$and $$\begin{aligned} L_{18}=5778=6666-888, \end{aligned}$$respectively. In...

From the well-known Fibonacci sequence, the number \(F_{10}=55=5\cdot 11\) is an example not only as a repdigit (a number with only one distinct digit) but also as a product of two repdigits with consecutive lengths, 5 and 11. Here we find all the Fibonacci numbers that can be written as the product of k repdigits with consecutive lengths.

In this paper, we prove that there are no positive integers a, b, c and d such that {Pa,Pb,Pc,Pd} is a Diophantine quadruple, where for a positive integer m, Pm is the mth Pell number.

We show that while the number of coprime compositions of a positive integer n into k parts can be expressed as a Q-linear combinations of the Jordan totient functions, this is never possible for the coprime partitions of n into k parts. We also show that the number pk′(n) of coprime partitions of n into k parts can be expressed as a C-linear combin...

Here, we show that if un = n2 n ±1, then the largest prime factor of un ± m! for n ≥ 0, m ≥ 2 tends to infinity with max{m, n}. In particular, the largest n participating in the equation un ± m! = 2 a 3 b 5 c 7 d with n ≥ 1, m ≥ 2 is n = 8 for which (8 · 2 8 + 1) − 4! = 3 4 · 5 2 .

We show that the k-generalized Fibonacci numbers that are concatenations of two repdigits have at most four digits.

In this paper, we show that if (Xn,Yn) is the nth solution of the Pell equation X2 − dY2 = ±1 for some non-square d, then given any integer c, the equation c = Xn − 2m has at most 2 integer solutions (n,m) with n ≥ 0 and m ≥ 0, except for the only pair (c,d) = (−1, 2). Moreover, we show that this bound is optimal. Additionally, we propose a conject...

It is conjectured that the sum $$ S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k} $$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in {\mathbb Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools...

Here we look at the Markov equations ax2+by2+cz2=dxyz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ax^2+by^2+cz^2=dxyz$$\end{document} with integer solutions (x, y, z...

Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$...

Let \(S=\{p_{1},\ldots ,p_{t}\}\) be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation \((F_n^{(k)})^s=p_{1}^{a_{1}}+\cdots +p_{t}^{a_{t}}\), in integer unknowns \(n\ge 1\), \(s\ge 1,~k\ge 2\) and \(a_i\ge 0\) for \(i=1,\ldots ,t\) such that \(\max \left\{ a_{i}: 1\le i\le t\right\...

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with four or more parameters. More precisely, consider $M$ a d-dimensional square matrix whose entries are rational functions in one or more real variables. Given initial and target vec...

A generalization of the well–known Fibonacci sequence is the ℓ-Fibonacci sequence (Fm(ℓ))m whose first ℓ terms are 0,…,0,1 and each term afterwards is the sum of the preceding ℓ terms. The k-Pell sequence (Pn(k))n, which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between the...

We show that if {Un}n≥0 is a Lucas sequence, then the largest n suc that |Un| = Cm1Cm2⋯Cmk with 1 ≤ m1 ≤ m2 ≤⋯ ≤ mk, where Cm is the mth Catalan number satisfies n < 6500. In case the roots of the Lucas sequence are real, we have n ∈{1, 2, 3, 4, 6, 8, 12}. As a consequence, we show that if {Xn}n≥1 is the sequence of the X coordinates of a Pell equa...

We prove Skolem's conjecture for the exponential Diophantine equation an+tbn=±cn under some assumptions on the integers a,b,c,t. In particular, our results together with Wiles' theorem imply that for fixed coprime integers a,b,c Fermat's equation an+bn=cn has no integer solution n≥3 modulo m for some modulus m depending only on a,b,c. We also provi...

In this paper, we find all the solutions of the title Diophantine equation in positive integers ( m , n , k , x ), where P i is the i th term of the Pell sequence.

Let {F n } n≥0 be the sequence of Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n+2 = F n+1 + F n for all n ≥ 0. In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ±4 which is a sum of two Fibonacci numbers, with a few exce...

In this paper, we give an algorithm which finds, for an integer base b≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\ge 2$$\end{document}, all squarefree integers...