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.
released 2026-09-24 | 8 theorems · 19 lemmas · 27 proofs · 29,024 words |
PLAY LEVEL 1 »(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 xδ.
released 2026-09-24 | 4 theorems · 22 lemmas · 21 proofs · 31,398 words |
PLAY LEVEL 2 »(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.
released 2026-09-17 | 4 theorems · 20 lemmas · 21 proofs · 16,981 words |
PLAY LEVEL 3 »(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 $\mu(p-1)=1$.