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OP-12112Status unassessedNumber theory

Difference Bases (AlphaEvolve problem 7)

Difference Bases (AlphaEvolve problem 7). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 11B13 · Additive bases, including sumsets

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
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Problem statement

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.

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Background

Repository result category: world record — AlphaEvolve improved on the best known result recorded at the time. See Section 6.3 in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

The repository page records no bounds for this problem.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Difference Bases (AlphaEvolve problem 7) linkApache-2.0 (repository license); problem pages reproduced with attribution.

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