Family 011Number theoryNo LeanNot yet digested

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.

Manuscripts in this family · 3

Weighted dilation graphs, smooth shifted primes and totient fibers

Sep 24, 2026·paper.pdf

We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than

n^{1-\varepsilon }

preimages. We also show that, for every fixed δ > 0, there are at least

x^{1-o(1)}

primes p in

2x\lt p\le5x

whose predecessors have no prime factor exceeding x

^{δ}

Full abstract

PDF on GitHub LaTeX source

The Poisson-Dirichlet law for prime predecessors

Sep 24, 2026·paper.pdf

For a prime p chosen uniformly from

3\le p\le x

, list the prime factors of

p-1

in decreasing order, with multiplicity. As

x\to\infty

, their logarithms, divided by

\log(p-1)

, converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.

Full abstract

PDF on GitHub LaTeX source

Prime Predecessors with an Even Number of Prime Factors

Sep 17, 2026·paper.pdf

We prove that there are infinitely many primes p for which

p-1

is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with

\mu(p-1)=1

Full abstract

PDF on GitHub LaTeX source

Prime-factor statistics of p−1p-1p−1

Manuscripts in this family · 3

Weighted dilation graphs, smooth shifted primes and totient fibers

The Poisson-Dirichlet law for prime predecessors

Prime Predecessors with an Even Number of Prime Factors

Prime-factor statistics of $p-1$