Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root
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arXiv
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| Format: | Preprint |
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2026
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| author | Godara, Vikas Sharma, Divyum |
| author_facet | Godara, Vikas Sharma, Divyum |
| contents | 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 $2,3,5,7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_28449 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root Godara, Vikas Sharma, Divyum Number Theory 11D61, 11J86, 11D72, 11B37 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 $2,3,5,7$. |
| title | Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root |
| topic | Number Theory 11D61, 11J86, 11D72, 11B37 |
| url | https://arxiv.org/abs/2605.28449 |