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arXiv · 2605.28449

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

Vikas Godara·Divyum Sharma
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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 $2,3,5,7$.

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Vikas Godara, Divyum Sharma. 2026-05-27. Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root. https://arxiv.org/abs/2605.28449

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