Erdős Problem #1148

Full solution by Ingo Althöfer, Przemek Chojecki and Wouter van Doorn with Gemini 3 Pro, Gemini 3.1 Pro, GPT-5.2 Pro, GPT-5.2 Thinking and GPT-5.4 Pro (16 Mar 2026). 2 AI contributions are recorded in all.

Problem
Erdős problem #1148 (number theory); current status: proved (Lean)
Credit
Ingo Althöfer, Przemek Chojecki and Wouter van Doorn with Gemini 3 Pro, Gemini 3.1 Pro, GPT-5.2 Pro, GPT-5.2 Thinking and GPT-5.4 Pro · AI working with humans
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-03-16 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
SOLVEDPer Tao's AI-contributions wiki

The problem

Can every large integer nn be written as n=x2+y2−z2n=x^2+y^2-z^2 with max⁡(x2,y2,z2)≤n\max(x^2,y^2,z^2)\leq n?

Statement quoted from erdosproblems.com/1148, maintained by Thomas Bloom. Original source: [Va99,1.25]. Current status there: proved (Lean).

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
24 Jan-16 Mar, 2026 AI working with humans Gemini 3 Pro, Gemini 3.1 Pro, GPT-5.2 Pro, GPT-5.2 Thinking, GPT-5.4 Pro Ingo Althöfer, Przemek Chojecki, Wouter van Doorn 🟢 Full solution
19 May, 2026 AI working with humans GPT-5.5 Pro Ingo Althöfer, Przemek Chojecki 🟢 Solution to stronger problem

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.

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