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PreprintCC BY 4.0arXiv

Coprime pairs with equal sum of divisors, assuming smooth shifted primes

  • Shisheng Li
  • Zhong Zhuang
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Abstract

Let h(x) be the number of pairs 1≤ a<b<x with gcd(a,b)=1 and σ(a)=σ(b). Erdős Problem #824 asks whether h(x)>x2−o(1). By an argument of Pollack and Pomerance, this follows if for every fixed δ>0 there are y1−o(1) primes p≤5y with p+1 free of prime factors exceeding yδ. 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)=x2−o(1), conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.

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Cite

BibTeXRIS
BibTeX
@misc{li2026coprime,
  title = {{Coprime pairs with equal sum of divisors, assuming smooth shifted primes}},
  author = {Shisheng Li and Zhong Zhuang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15674},
  eprint = {2610.15674},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15674},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Coprime pairs with equal sum of divisors, assuming smooth shifted primes
AU  - Shisheng Li
AU  - Zhong Zhuang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15674
UR  - https://arxiv.org/abs/2610.15674
AB  - Let h(x) be the number of pairs 1≤ a<b<x with gcd(a,b)=1 and σ(a)=σ(b). Erdős Problem #824 asks whether h(x)>x^2−o(1). By an argument of Pollack and Pomerance, this follows if for every fixed δ>0 there are y^1−o(1) primes p≤5y with p+1 free of prime factors exceeding y^δ. 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.
ER  -