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Erdős Problem 416: does V(2x)/V(x)→2V(2x)/V(x)\to2 for the number V(x)V(x) of totient values up to xx, and is there an asymptotic formula?

Erdős problem #416 · erdosproblems.com/416

Let V(x)V(x) count the integers v≤xv\le x for which φ(m)=v\varphi(m)=v is solvable. Pillai showed these values have density zero; after Erdős, Erdős-Hall, Pomerance and Maier-Pomerance, Ford (1998) determined V(x)V(x) up to a bounded factor and showed V(cx)−V(x)≍cV(x)V(cx)-V(x)\asymp_c V(x), but this falls short of an asymptotic formula. Erdős and Hall asked whether V(cx)/V(x)→cV(cx)/V(x)\to c for fixed c>1c>1. Does V(2x)/V(x)→2V(2x)/V(x)\to2, and is there an asymptotic formula for V(x)V(x)?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Multiplicative number theory; values of arithmetic functions
Posed by
Paul Erdős and R. R. Hall (1976, Mathematika); repeated by Erdős (1979)
Year posed
1976
Years open
50y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
18 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The paper gives an explicit asymptotic equivalent V(x)∼xlog⁡xG(x)A(θ(x))V(x)\sim\frac{x}{\log x}G(x)A(\theta(x)), where the bounded positive coefficient AA depends on a phase θ(x)∈[0,1)\theta(x)\in[0,1) and is a uniform limit of functions defined from finite prime and integer data, and proves V(cx)/V(x)→cV(cx)/V(x)\to c for every fixed c>0c>0, answering Erdős-Hall. It also gives asymptotics for totients whose least preimage lies in (kx,(k+1)x](kx,(k+1)x], a further question of Erdős (1979), with a positive coefficient for k=1,2k=1,2 and identically zero counts when no totient dd has least preimage above kdkd. Not shown: a closed-form constant (the coefficient is a limit, not a number), or secondary terms.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result 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. Single manuscript dated September 25, 2026, building on Ford's structure theorems for totient preimages.

Verification

No independent mathematician has checked this yet. Checked here: introduction read against Erdős Problem 416 and Erdős-Hall's question. Lean: the challenge lean/ComparatorChallenges/TotientAsymptotic.json (solution_module OAI.NumberTheory.TotientAsymptotic.UnconditionalMain, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statement totient_asymptotic_formula was read here: with VV defined as the number of v∈[1,x]v\in[1,x] that are totients, it asserts V(x)/mainTerm(x)→1V(x)/\mathrm{mainTerm}(x)\to1 for an explicit main term built from finite arithmetic approximants (with uniform convergence of the approximants and two-sided bounds on the coefficient) and V(cx)/V(x)→cV(cx)/V(x)\to c for every c>0c>0. That is the headline. Not rebuilt here.

Sources

Changelog1 change

Discussion