Universal sums of generalized polygonal numbers of almost prime length
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914461154017280 |
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| author | Banerjee, Soumyarup Kane, Ben Ng, Kwan To |
| author_facet | Banerjee, Soumyarup Kane, Ben Ng, Kwan To |
| contents | In this paper, we consider universal sums of generalized polygonal numbers. Fixing $m\in\mathbb{N}_{\geq 3}$, we show two finiteness theorems for universal sums of generalized polygonal numbers whose inputs have a restricted number $L$ of prime divisors (counting multiplicity) away from an finite set of exceptional primes. In the first theorem, we fix $m$ and uniformly bound the finite check independent of $L\geq 900$, and in the second theorem, we give an optimal bound for the finiteness check if $L$ is larger than a constant times $\log(m)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07826 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universal sums of generalized polygonal numbers of almost prime length Banerjee, Soumyarup Kane, Ben Ng, Kwan To Number Theory 11F11, 11F27, 11F30, 11E20, 11E45, 11N36 In this paper, we consider universal sums of generalized polygonal numbers. Fixing $m\in\mathbb{N}_{\geq 3}$, we show two finiteness theorems for universal sums of generalized polygonal numbers whose inputs have a restricted number $L$ of prime divisors (counting multiplicity) away from an finite set of exceptional primes. In the first theorem, we fix $m$ and uniformly bound the finite check independent of $L\geq 900$, and in the second theorem, we give an optimal bound for the finiteness check if $L$ is larger than a constant times $\log(m)$. |
| title | Universal sums of generalized polygonal numbers of almost prime length |
| topic | Number Theory 11F11, 11F27, 11F30, 11E20, 11E45, 11N36 |
| url | https://arxiv.org/abs/2604.07826 |