Difference Bases

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open.

Problem
Problem 7 in the AlphaEvolve repository: Difference Bases
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

For any natural number nn, let Δ(n)\Delta(n) be the size of the smallest set BB of integers such that every natural number from 11 to nn is expressible as a difference of two elements of BB (such sets are known as difference bases for the interval {1,…,n}\{1,\dots,n\}). Write C(n)≔Δ2(n)/nC(n) \coloneqq \Delta^2(n)/n, and C≔inf⁡n≥1C(n)C \coloneqq \inf_{n \geq 1} C(n). Establish upper and lower bounds on CC that are as strong as possible.

Statement quoted from problem 7 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.3 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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