Short proofs in combinatorics and number theory

TL;DR

This paper presents three concise proofs addressing questions by Erdős in combinatorics and number theory, involving prime factors, additive bases, and distribution of fractional parts.

Contribution

It introduces novel short proofs for three classical problems in combinatorics and number theory, expanding understanding of these topics.

Findings

Proved small prime factors of binomial coefficients.

Showed additive bases can be split into parts with bounded gaps.

Established well-distribution of fractional parts of scaled primes.

Abstract

We give a triplet of short proofs, each of which answers a question raised by Erd\H{o}s. 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.