A Complete Answer to Erd\H{o}s Problem 690

Abstract

Let \(d_k(p)\) denote the natural density of positive integers whose \(k\)-th smallest prime divisor is \(p\). Erd\H{o}s asked whether, for each fixed \(k\), the sequence \(p\mapsto d_k(p)\) is unimodal as \(p\) ranges over the primes. Cambie proved that unimodality holds for \(1\le k\le3\) and verified non-unimodality for \(4\le k\le20\). We prove that \(p\mapsto d_k(p)\) is not unimodal for every \(k\ge4\), completing the classification. An exact first-difference criterion reduces the problem to comparing a symmetric-polynomial ratio with prime gaps. Explicit estimates for prime-counting functions, certified finite computations, one certified large prime gap, one certified twin prime, and a uniform Chinese-remainder construction then produce, for every \(k\ge4\), a strict descent followed by a later strict ascent.