Number theory · Oct 6, 2026

Prime-factor statistics of $p-1$

Proves that the normalized ordered logarithms of the prime factors of

p-1

, counted with multiplicity, converge jointly to the Poisson–Dirichlet law

\mathrm{PD}(1)

as p ranges uniformly over primes up to x and

x\to\infty

. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than

n^{1-\varepsilon}

totient preimages, for every ε > 0.

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