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A Resolution of Erd\H{o}s Problem 768: the Sylow Divisor Condition

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Eric Li (Trinity College, University of Cambridge)

Number Theory · Submitted 23 Jun 2026 · 12 references

Pith's verdict: Exact constant for Erdős 768 is proved two-sided; the written argument holds and the Lean claim is the only unreproducible bit.

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arXiv ID2606.24872 (version 2)
Pith IDLKEJW6XE, Pith's permanent record of this paper and its review
CitationEric Li (Trinity College, University of Cambridge). "A Resolution of Erd\H{o}s Problem 768: the Sylow Divisor Condition." arXiv:2606.24872 (2026). https://pith.science/paper/LKEJW6XE
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The paper

The paper claims

The density of integers with a Sylow-style witness divisor for every prime factor decays as exp(−(1/(2√log 2)+o(1))√(log x) log log x).

The reading

Erdős asked how rare the positive integers n are that, for every prime p dividing n, admit some divisor d>1 of n with d≡1 mod p. This paper proves that the count A(x) of such n≤x satisfies log(x/A(x)) divided by √(log x) log log x tends exactly to 1/(2√log 2). The same constant appears from a constructive lower bound that multiplies one prime from each of r carefully spaced logarithmic intervals, after a large-sieve cleaning step that makes subset products hit 1 mod each chosen prime, and from an upper bound that compresses n by a deterministic square-free factor built from canonical witnesses and then bounds the fibers by growing divisor moments. The matching rates settle the leading constant that Erdős left open and give a sharp elementary upper bound on possible orders of nonabelian finite simple groups.

What carries the argument

The matching of two exponential rates: a fourth-moment large-sieve cleaning of logarithmic prime layers plus a subset-product second-moment lemma for the lower bound, and a deterministic compression map n↦n/Q(n) whose fibers are injectively reconstructed from a half-row record and then bounded by growing divisor moments for the upper bound; optimizing max{λ/2, λ/4 + 1/(4λ log 2)} yields the constant.

Extended reading notes

Core claim

The limit lim log(x/A(x)) / (√(log x) log log x) equals 1/(2√log 2), so A(x)=x exp(−(1/(2√log 2)+o(1))√(log x) log log x). Both the lower-bound construction from cleaned logarithmic prime layers and the upper-bound compression by canonical witnesses optimize to this same constant.

Load-bearing premise

After the large sieve is applied to products of primes from each logarithmic layer, only an exponentially small set of target primes can still have a large Fourier coefficient; without that cleaning the constructive lower bound cannot reach the matching constant.

Original abstract

We resolve Erd\H{o}s Problem 768. Let $A(x)$ count the positive integers $n\le x$ such that, for every prime $p\mid n$, there is a divisor $d>1$ of $n$ with $d\equiv 1 \pmod p$. Erd\H{o}s asked whether $A(x)/x=\exp(-(c+o(1))\sqrt{\log x}\log\log x)$ for some constant $c>0$. We prove that this holds with $c=1/(2\sqrt{\log 2})$; equivalently, $\log(x/A(x))/(\sqrt{\log x}\log\log x)$ tends to $1/(2\sqrt{\log 2})$. The lower bound is obtained from primes in disjoint logarithmic intervals using a fourth-moment argument based on the multiplicative large sieve and a subset-product second moment. The upper bound uses canonical witness divisors, a deterministic compression map, an injective reconstruction theorem for its fibers, and growing divisor moments. Thus the paper determines the exact leading constant in Erd\H{o}s Problem 768. The main theorem and its complete proof have been formally verified in the Lean 4 proof assistant.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites as background 1 paper
  • Related random subset-sum questions in a different full-coverage model have recently been studied by Ma and Tang [ref] .

    2602.05768 · Introduction · 0.98

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