VibeMathedMath problems solved by AI
All problems

Erdős Problem #856

Erdős problem #856 · erdosproblems.com/856

Let k≥3k\geq 3 and fk(N)f_k(N) be the maximum of ∑n∈A1n\sum_{n\in A}\frac{1}{n} over all A⊆{1,…,N}A\subseteq\{1,\ldots,N\} containing no kk subsets with the same pairwise least common multiple. Estimate fk(N)f_k(N). The claimed answer: fk(N)=(log⁡N)γk+o(1)f_k(N)=(\log N)^{\gamma_k+o(1)}, where γk\gamma_k is a weighted generalization of the Tang-Zhang sunflower capacity.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Argument
Field
Number Theory
Posed by
Paul Erdős
Year posed
1970
Years open
56y
Solved
2026-04-15
Model
GPT-5.4 Pro
Vendor
OpenAI
Collaborators
Przemysław Chojecki
Verification
Unreviewed
Publication
Announced
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance

What the AI did

A weighted version of the Tang-Zhang sunflower-capacity argument giving the exact logarithmic exponent was developed with GPT-5.4 Pro, using a mass-transport idea from the forum's discussion of problem #1196.

Verification

An AI screening found no issues and no prior literature with the result; the site's owner unpacked and restated the main claim without checking details, a Lean formalization attempt hit missing mathlib prerequisites, and the problem is still listed open.

Sources

erdosproblems.com/856

Discussion