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Short proofs in combinatorics and number theory

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, 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 $\{αp\}$ 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.

Short proofs in combinatorics and number theory

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 , the second concerns whether an additive basis can always be split into pieces and such that each of has bounded gaps, and the final concerns whether 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.

Paper Structure

This paper contains 4 sections, 6 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Thresholds for binomial coefficients
  3. Basis of order two with no syndetic splits
  4. Fractional parts of $\{\alpha p\}$

Key Result

Theorem 2.1

For $n$ sufficiently large, we have that Furthermore, there exists a sequence $n_j\to \infty$ such that

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of \ref{['thm:basis-main']}
  • Theorem 4.1
  • Theorem 4.2
  • ...and 1 more