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Erdős's totient-fiber conjecture: φ(m)=n\varphi(m)=n has more than n1−εn^{1-\varepsilon} solutions for infinitely many nn

Erdős problem #821 · erdosproblems.com/821

Let g(n)=#{m≥1:φ(m)=n}g(n)=\#\{m\ge1:\varphi(m)=n\} count the preimages of nn under Euler's totient function. An elementary bound gives g(n)≪ηn1+ηg(n)\ll_\eta n^{1+\eta}; Pillai showed lim sup⁡g(n)=∞\limsup g(n)=\infty and Erdős (1935) showed g(n)>ncg(n)>n^c infinitely often for some c>0c>0. Pomerance (1980) formulated the question as whether the supremum CC of exponents with g(n)>ncg(n)>n^c infinitely often equals 11, attributing it to Erdős, and showed it would follow from enough primes pp whose predecessor p−1p-1 has no prime factor above pεp^\varepsilon. The best exponent known was 0.71570.7157 (Lichtman 2022), after Baker-Harman's 0.70390.7039. Is it true that for every ε>0\varepsilon>0 there are infinitely many nn with g(n)>n1−εg(n)>n^{1-\varepsilon}?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Multiplicative number theory; Euler's totient function
Posed by
Paul Erdős (1956, as attributed by Pomerance 1980); Erdős Problem #821
Year posed
1956
Years open
70y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every ε>0\varepsilon>0 there are infinitely many nn with g(n)>n1−εg(n)>n^{1-\varepsilon}, which is optimal in the exponent because g(n)≪ηn1+ηg(n)\ll_\eta n^{1+\eta}. The input is Theorem 1.2: for fixed 0<δ<1/40<\delta<1/4, at least x1−o(1)x^{1-o(1)} primes p∈(2x,5x]p\in(2x,5x] have P+(p−1)≤xδP^+(p-1)\le x^\delta; in particular there are infinitely many primes with P+(p−1)≤pεP^+(p-1)\le p^\varepsilon for every ε>0\varepsilon>0. It does not give a positive proportion of such primes (that stronger statement is in the Poisson-Dirichlet companion), and the o(1)o(1) is not made explicit.

What the AI did

The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This manuscript proves the smooth shifted-prime count and the totient consequence; its graph and kernel estimates are reused by the Poisson-Dirichlet companion (separate entry for the Ford-Konyagin-Luca conjecture).

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 and the introduction were read against the conjecture as Pomerance formulates it and as erdosproblems.com states Problem #821. No Lean formalization exists for this family. The deduction from smooth shifted primes to large fibers is the classical Erdős-Pomerance argument (Pomerance's Theorem B), written out in full; the new input is the x1−o(1)x^{1-o(1)} count of primes with xδx^\delta-smooth predecessors, which goes far beyond the previous exponent 0.2843 and should be treated as unverified until specialists examine it. Not refereed here.

Sources

Changelog1 change

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