On Additive Representations of Integers by Binomial Coefficients
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
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| _version_ | 1866910172207644672 |
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| author | Povolotsky, Alexander |
| author_facet | Povolotsky, Alexander |
| contents | For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional cases, general existence results have received less explicit treatment.
This paper provides: $\bullet$ explicit elementary proofs for the cases ($k=2$) and ($k=3$), $\bullet$ a comparison with classical polygonal number theory, $\bullet$ an explanation of why naive counting arguments fail for general ($k$), $\bullet$ conditional and unconditional existence results for general ($k$), $\bullet$ and a discussion of quantitative bounds and computational evidence.
Together these give a unified and transparent framework for understanding additive representations by binomial coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24828 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Additive Representations of Integers by Binomial Coefficients Povolotsky, Alexander Combinatorics For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional cases, general existence results have received less explicit treatment. This paper provides: $\bullet$ explicit elementary proofs for the cases ($k=2$) and ($k=3$), $\bullet$ a comparison with classical polygonal number theory, $\bullet$ an explanation of why naive counting arguments fail for general ($k$), $\bullet$ conditional and unconditional existence results for general ($k$), $\bullet$ and a discussion of quantitative bounds and computational evidence. Together these give a unified and transparent framework for understanding additive representations by binomial coefficients. |
| title | On Additive Representations of Integers by Binomial Coefficients |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.24828 |