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

2024/09/03 by Ibrahimov, Seyran S., Mahmudov, Nazim I.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2409.02047

Abstract

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 ≥ 1 and k ≥ 3. Here, \ F_t \_t=0^∞ denotes the Fibonacci numbers, defined by the recurrence relation F₀ = 0, F₁ = 1, and F_t = F_t-1 + F_t-2 for t ≥ 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.

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