On a Variant of Pillai's problem involving convergent denominators of quadratic irrationals

2025/08/15 by Mittal, Mohit
#11D45 #11D61 #11J86 #11R04 #11R11 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.11243

Abstract

Let (q_{α, n})_{n ≥ 0} be the sequence of convergent denominators to the simple continued fraction expansion of α. For certain specific choices of α, this sequence is a Lehmer sequence. In this paper, we show that there are only finitely many integers c such that the equation q_{α, n} - q_{β, m} = c has at least two distinct solutions (n,m), where α,β are quadratic irrationals with ℚ(α)≠ ℚ(β). In specific instances, we solve the equation q_{α, n} - q_{β, m} = c completely and explicitly list all solutions.

Citations

• On the problem of Pillai with Fibonacci numbers, Padovan numbers, and Tribonacci numbers and powers of 3
• On a variant of Pillai's problem II
• Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers
• A generalization of a theorem of Baker and Davenport

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