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Erdős Problem 768 solved with exact constant 1/(2√log 2)
The Sylow Divisor Condition: a Resolution of ErdH{o}s Problem 768
The density of integers where every prime factor p has a divisor ≡1 mod p is exp(-(c+o(1))√logx loglogx) with c=1/(2√log2).
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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.
