Erdős Problem #822

GPT-5 found that the problem had already been solved in the literature (13 Oct 2025).

Problem
Erdős problem #822 (number theory); current status: proved
Credit
GPT-5 · Literature search
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2025-10-13 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PRIOR
ARTPer Tao's AI-contributions wiki

The problem

Does the set of integers of the form n+ϕ(n)n+\phi(n) have positive (lower) density?

Statement quoted from erdosproblems.com/822, maintained by Thomas Bloom. Original source: [Er74b]. Current status there: proved.

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 Outcome
13 Oct, 2025 Literature search GPT-5 🟢 Full solution found

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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