Short proofs in combinatorics and number theory

by Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant

Abstract

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 \(\{\alpha p\}\) 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.

Approximate citation

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant
Short proofs in combinatorics and number theory

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