Erdős Problem #349

Partial progress by GPT-5.2 Thinking (9 Mar 2026): improved partial result.

Problem
Erdős problem #349 (number theory, complete sequences); current status: open
Credit
GPT-5.2 Thinking · AI building on known literature
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-03-09 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PARTIALPer Tao's AI-contributions wiki

The problem

For what values of t,α∈(0,∞)t,\alpha \in (0,\infty) is the sequence ⌊tαn⌋\lfloor t\alpha^n\rfloor complete (that is, all sufficiently large integers are the sum of distinct integers of the form ⌊tαn⌋\lfloor t\alpha^n\rfloor)?

Statement quoted from erdosproblems.com/349, maintained by Thomas Bloom. Original source: [ErGr80,p.57]. 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 Outcome Literature
9 Mar, 2026 AI building on known literature GPT-5.2 Thinking 🟡 Improved partial result 🟡 van Doorn (2026)

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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