Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root

Vikas Godara; Divyum Sharma

paper

Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root

arXiv:2605.28449

Abstract

Let $C_{n} = n 2^{n} + 1$ denote the $n$ th Cullen number. There has been recent interest in finding all Cullen numbers having a given Diophantine property. We prove that, for a fixed integer $k$ and bounded integers $a_{1}, \dots, a_{k}$ , the greatest prime divisor of $C_{n} - a_{1} m_{1}! - \dots - a_{k} m_{k}!$ tends to infinity, in an effective way. We prove this for some more general families of ternary recurrence sequences as well. We also solve the Diophantine equation $C_{n} = m_{1}! + m_{2}! + s,$ where $s$ is a positive integer composed of primes $2, 3, 5, 7$ .