Erdős Problem #689

A claim by Boris Alexeev, Przemek Chojecki, Dogmachine, jleng01, Mehtaab Sawhney, Terence Tao and Malek Zribi with Codex, GPT-5.2 and GPT-5.5 Pro (29 Oct 2025) that has not yet been verified: candidate full solution.

Problem
Erdős problem #689 (number theory); current status: open
Credit
Boris Alexeev, Przemek Chojecki, Dogmachine, jleng01, Mehtaab Sawhney, Terence Tao and Malek Zribi with Codex, GPT-5.2 and GPT-5.5 Pro · AI working with humans
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2025-10-29 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
UN-
VERIFIEDPer Tao's AI-contributions wiki

The problem

Let nn be sufficiently large. Is there some choice of congruence class apa_p for all primes 2≤p≤n2\leq p\leq n such that every integer in [1,n][1,n] satisfies at least two of the congruences ≡ap(modp)\equiv a_p\pmod{p}?

Statement quoted from erdosproblems.com/689, maintained by Thomas Bloom. Original source: [Er79d][Er80,p.108]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Humans Outcome
29 Oct, 2025-2 Jun, 2026 AI working with humans Codex, GPT-5.2, GPT-5.5 Pro Boris Alexeev, Przemek Chojecki, Dogmachine, jleng01, Mehtaab Sawhney, Terence Tao, Malek Zribi ⚪ Candidate full solution

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎