Short proofs in combinatorics and number theory
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912997474041856 |
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| author | Alexeev, Boris Putterman, Moe Sawhney, Mehtaab Sellke, Mark Valiant, Gregory |
| author_facet | Alexeev, Boris Putterman, Moe Sawhney, Mehtaab Sellke, Mark Valiant, Gregory |
| contents | We give a triplet of short proofs, each of which answers a question raised by Erdős. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i + A_i$ has bounded gaps, and the final concerns whether $\{αp\}$ is "well-distributed" in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29961 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Short proofs in combinatorics and number theory Alexeev, Boris Putterman, Moe Sawhney, Mehtaab Sellke, Mark Valiant, Gregory Combinatorics Number Theory We give a triplet of short proofs, each of which answers a question raised by Erdős. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i + A_i$ has bounded gaps, and the final concerns whether $\{αp\}$ is "well-distributed" in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI. |
| title | Short proofs in combinatorics and number theory |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2603.29961 |