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

Abstract

Let $C_n=n2^n+1$ 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 $a_1,\ldots,a_k$, the greatest prime divisor of $C_n-a_1m_1!-\cdots-a_km_k!$ 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 $2,3,5,7$.