arXiv subjects

Seyran S. Ibrahimov

Publications and source records attributed to Seyran S. Ibrahimov.

3 recordsLinked to original sources

arXiv2026

On a Diophantine Equation Involving Lucas Numbers

Let L_t denote the t-th Lucas number. We prove that the Diophantine equation L_m^{n+k} + L_m^n = L_r has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof.

math.NT↗

arXiv2024

On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$

In this paper, we examine the Diophantine problem given by the equation $F_n = F_l^k (F_l^m - 1)$, where $n, l, m \geq 1$ and $k \geq 3$. Here, $\{ F_t \}_{t=0}^{\infty} $ denotes the Fibonacci numbers, defined by the recurrence relation $F_0 = 0$, $F_1 = 1$, and $F_t = F_{t-1} + F_{t-2}$ for $t \geq 2$. By applying Matveev's theorem, which provides lower bounds for linear forms in logarithms of algebraic numbers, along with a modified Baker-Davenport reduction method and a divisibility property of Fibonacci numbers, we show that $(n, l, k, m) = (6, 3, 3, 1)$ is the only positive integer quadruple that satisfies this equation.

math.NT↗

arXiv2024

A Pillai-Catalan-type problem involving Fibonacci numbers

This paper addresses A Pillai-Catalan-type problem assosiated with Fibonacci numbers. Let $F_{n}$ be the Fibonacci numbers defined by the recurrence relation $F_{1}=1$, $F_{2}=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for all $n\geq 3$. We will find all positive integer solution to the equation $3^{x}-F_{n}2^{y}=1$ using properties of Fibonacci numbers, linear forms of logarithms, and the Baker-Davenport reduction method.

math.NT↗