REVIEW
Coprime pairs with equal sum of divisors, assuming smooth shifted primes
Number Theory · Submitted 11 Oct 2026 · 10 references
Pith's verdict: Honest conditional result with a machine-checked implication, but the main conclusion rides entirely on an unrefereed preprint the authors did not verify.
T0 · AI review
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Referee report
The paper
The paper claims
Erdős Problem #824 conditionally resolved if unrefereed preprint's smooth-prime estimates hold.
The reading
What carries the argument
The key machinery is the combinatorial pigeonhole argument of Pollack and Pomerance, which the paper rewrites with explicit bounds (parameters B, η, δ, and k). From a set P of primes with smooth successor, one forms A, the set of all products of k distinct primes from P. Every σ(a) is then a smooth integer, so the range of σ on A is bounded by U, the count of smooth integers ≤x. By Cauchy–Schwarz and careful counting, one shows that many distinct a∈A share the same σ-value. Removing a small common prime divisor yields many coprime pairs with equal σ. The paper also checks a technical point in the OpenAI preprint: the determinant parametrization in the Type II estimate works for p+1 after a sign change, because the shift-cancellation theorem applies to both signs of the shift k.
Extended reading notes
Core claim
The paper's central contribution is twofold. First, it makes explicit the argument (due to Pollack and Pomerance) that Hypothesis A+—the existence of $y^{{1-o(1)}}$ primes p≤5y with p+1 free of prime factors exceeding y^δ for every δ>0—implies h(x)>$x^{{2-ε}}$ for every ε>0, with the pairs squarefree. This implication is written out with explicit constants and formalized in Lean 4, providing an unconditional machine-checked proof of the conditional statement. Second, the paper checks that the OpenAI preprint's construction of smooth shifted primes for p−1 extends to p+1 by a sign change in two sections. On that basis, it states that if the preprint's core estimates (about 2800 lines, unverified) are correct, then Hypothesis A+ holds and consequently h(x)=$x^{{2-o(1)}}$. The authors do not vouch for the preprint and emphasize that its main new method remains unverified.
Load-bearing premise
The entire conditional result depends on the correctness of an unrefereed OpenAI preprint whose core estimates (Sections 3–5, about 2800 lines) the authors have not independently verified.
Editorial extensions
If this is right
- If the OpenAI preprint's estimates are correct, Erdős Problem #824 is resolved: h(x)=x^{2-o(1)}.
- The Lean formalization proves the implication unconditionally, so any future proof of Hypothesis A+ automatically yields the same bound for h(x).
- The pairs may be taken squarefree, a refinement that follows from the argument.
- The explicit parametrization highlights how large the set of smooth shifted primes must be to achieve the exponent 2−ε.
Reading between the lines
- The paper's conditional resolution underscores the growing role of AI-generated mathematics in open problems; the burden of verification now rests on the community to either validate or refute the OpenAI preprint's core estimates.
- If the preprint's method is verified, it could likely be extended to other shifted primes (e.g., p+a for fixed a) or to Euler's totient function φ, where analogous problems exist.
- The Lean formalization, while covering only the combinatorial reduction, may serve as a reusable framework for future conditional results in smooth-prime contexts.
Original abstract
Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Erd\H{o}s Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\delta$. We write this implication out with explicit parameters and formalize it in Lean 4. We then check that the argument of the recent OpenAI preprint "Weighted dilation graphs, smooth shifted primes and totient fibers", which claims the corresponding statement for $p-1$, applies to $p+1$ after a change of sign in two of its sections. Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Erd\H{o}s Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\delta$. We write this implication out with explicit parameters and formalize it in Lean 4. We then check that the argument of the recent OpenAI preprint "Weighted dilation graphs, smooth shifted primes and totient fibers", which claims the corresponding statement for $p-1$, applies to $p+1$ after a change of sign in two of its sections. Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. | background 0.98 | 0.15 | 0.03 | asserted 0.55 |
| s1 | Erd\H{o}s Problem #824 asks whether $h(x)>x^{2-o(1)}$. | background 0.99 | 0.04 | 0.05 | asserted 0.37 |
| s2 | By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\delta$. | background 0.75 | 0.08 | 0.11 | asserted 0.96 |
| s3 | We write this implication out with explicit parameters and formalize it in Lean 4. | method 0.58 | 0.93 | 0.03 | asserted 0.92 |
| s4 | We then check that the argument of the recent OpenAI preprint "Weighted dilation graphs, smooth shifted primes and totient fibers", which claims the corresponding statement for $p-1$, applies to $p+1$ after a change of sign in two of its sections. | method 0.72 | 0.82 | 0.04 | asserted 0.98 |
| s5 | Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified. | limitation 0.68 | 0.52 | 0.90 | hedged possibility 0.51 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
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