Related papers: On Certain Diophantine Equations Involving Lucas N…

In this work we develop exact formulas to the number of solutions of $ax+by+cz=n$ in some special cases. In 2020, Binner gave a formula for the number of non negative integer solutions, $N(a,b,c;n)$ in non-negative integer pairs $(x,y,z),$…

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of…

In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.

Let $k,l\geq2$ be fixed integers. In this paper, firstly, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(lx)^{k}=y^{n}$ in integers $x,y,n$ with $x,y\geq1, n\geq2$ satisfy $n<C_{1}$ where $C_{1}=C_{1}(l,k)$ is an…

We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell…

The subject matter of this work is the diophantine equation x^n+y^m=c(x^k)(y^l), where n,m,k,l,c are natural numbers.We investigate this equation from the point of view of positive integer solutions.A preliminary examination of sources such…

In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport…

Balancing numbers $n$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+\cdots+(n+r)$, where $r$ is called the balancer corresponding to the balancing number $n$. By slightly modifying, $n$ is the…

We show that for the classical Fibonacci sequence (Fn) and the Lucas sequence (Ln) the following identity holds for every integer n >= 2: (n-1)Fn equals the sum from k=1 to n-1 of Lk multiplied by F(n-k). Equivalently, this gives a…

A composite positive integer n is Lehmer if \phi(n) divides n-1, where \phi(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no…

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers.…

We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by $\{F_n(\th)\}$ and $\{L_n(\th)\}$, respectively. We evaluate the Hankel determinants with entries $\{1/F_{j+k+1}(\th): 0\le i,j \le n\}$ and…

In [1] it is shown that the Diophantine equation $(k!)^n+k^n=(n!)^k+n^k$ only has the trivial solution $n=k$, and $(k!)^n-k^n=(n!)^k-n^k$ only has the solutions $n=k$, $(n, k)=(1, 2),$ and $(2, 1)$. In this article we find all solutions of…

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1…

We explicitly solve the diophantine equations of the form $$ A_{n_1}A_{n_2}\cdots A_{n_k}\pm 1 = B_m^2 $$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D.…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$…

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we…

Let $(U_{n}(P,Q) $ and $(V_{n}(P,Q) $ denote the generalized Fibonacci and Lucas sequence, respectively. In this study, we assume that $Q=1.$ We determine all indices $n$ such that $U_{n}=5\square $ and $U_{n}=5U_{m}\square $ under some…