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Erdős Problem #1201

Erdős problem #1201 · erdosproblems.com/1201

Is it true that for every ϵ,η>0\epsilon,\eta>0 there exists a kk such that the density of nn for which P(n(n+1)⋯(n+k))>n1−ϵP(n(n+1)\cdots(n+k))>n^{1-\epsilon} is at least 1−η1-\eta, where P(m)P(m) is the greatest prime divisor of mm? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.

Result
Proved(see note)
Status
Partial result
AI contribution
AI co-developed
Method
Argument
Field
Number Theory, Primes
Posed by
Paul Erdős
Year posed
1976
Years open
50y
Solved
2026-04-30
Model
GPT-5.5 Pro
Vendor
OpenAI
Collaborators
Przemysław Chojecki
Verification
Unreviewed
Publication
Announced
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open

What the AI did

The deduction from the Matomäki-Radziwiłł theorem on multiplicative functions was written by GPT-5.5 Pro; Tao and Sawin's forum discussion pinned down exactly what the known results do and do not give for this problem.

Verification

Discussed on the problem's forum, including by Tao, who delineated the remaining natural-density gap; no independent review of the note itself.

Sources

erdosproblems.com/1201

Discussion