Prime-factor statistics of p-1

1 · The essentials

What happens when you subtract one from a prime number and split the result into prime factors? This family of manuscripts studies the sizes and number of those factors. To compare their sizes, it takes their logarithms and divides them by the logarithm of the number being factored. The second manuscript claims that, when a prime is chosen at random below an increasingly large bound, the resulting shares approach a specific probability law. The result covers several of the largest shares considered together, not just the largest one. The companion manuscripts announce results about cases where all the factors are small, certain factorizations without repetitions, and how many inputs give the same value of Euler’s totient function. These results are presented in the manuscripts; the main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

An imperfect image is a bar cut into pieces, with the pieces then arranged from largest to smallest. Each piece here represents the share of the total logarithm contributed by one prime factor, and the second manuscript describes the limiting pattern of these sizes. The image does not capture the arithmetic restrictions on which cuts are possible.

Why it matters

The second manuscript presents its result as a resolution of the Ford–Konyagin–Luca conjecture. It describes the relative sizes of large factors of $p - 1$ together, rather than considering just one factor. The main manuscript also announces a result on large fibers of Euler’s totient function, presented as a resolution of a conjecture of Erdős.

2 · Going further

Choose a prime $p$ uniformly among those with $3 \leq p \leq x$ , and list the prime factors of $p - 1$ in decreasing order, retaining repetitions. Writing these factors as $q_{1} \geq q_{2} \geq \dots$ , the quantities of interest are $lo g q_{i} / lo g (p - 1)$ . The second manuscript claims that their finite joint distributions converge to those of the Poisson–Dirichlet law with parameter one, written $PD (1)$ , as $x \to \infty$ . In other words, convergence covers any fixed number of positions in this ordered list.

The main manuscript announces two complementary results: for every $ε > 0$ , infinitely many positive integers $n$ have more than $n^{1 - ε}$ preimages under Euler’s totient function; and, for every fixed $δ > 0$ , there are at least $x^{1 - o (1)}$ primes in $2 x < p \leq 5 x$ whose predecessors have no prime factor exceeding $x^{δ}$ . Its title highlights weighted dilation graphs, but the abstract does not explain their role in the argument. The supplied abstracts therefore do not support an account of the argument’s mechanism or establish logical dependencies between the manuscripts.

The third manuscript announces infinitely many primes $p$ for which $p - 1$ is squarefree and has an even number of prime factors. It gives the equivalent formulation $μ (p - 1) = 1$ . Thus the three texts address related but distinct aspects: relative factor sizes, the existence of predecessors with small factors, and a parity constraint without repeated factors.

Useful notions

Prime numbers and factorization
Logarithms
Uniform sampling and convergence in distribution
Euler’s totient function
Squarefree integers and the Möbius function

What was known before / what is new

Before

The source identifies a conjecture of Ford, Konyagin and Luca concerning the limiting law of prime factors of prime predecessors. It also mentions a conjecture of Erdős on the largest fibers of Euler’s totient function. It supplies neither a detailed history nor earlier partial results.

New

The second manuscript announces finite joint convergence of normalized factor logarithms to $PD (1)$ . The main manuscript announces bounds on large totient fibers and on the abundance of prime predecessors with small factors. The third announces infinitely many squarefree prime predecessors with an even number of prime factors.

Limits and assumptions

The limiting law concerns a prime chosen uniformly from $3 \leq p \leq x$ as $x$ tends to infinity, with factors counted with multiplicity; the stated convergence is in every finite joint distribution. The small-factor result requires fixed $δ > 0$ and concerns $2 x < p \leq 5 x$ . The preimage result holds for every $ε > 0$ and infinitely many $n$ , not for all $n$ . The abstracts provide neither a convergence rate nor details of the arguments, and do not specify remaining open questions. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Multiplicity: Counting factors with multiplicity means retaining every repetition of the same prime factor.
Finite joint distribution: This is the probability law describing a fixed number of quantities together, here logarithmic shares at specified positions.
Poisson–Dirichlet law with parameter one: Written $PD (1)$ , this is the limiting law for ordered logarithmic shares announced in the second manuscript.
Fiber of Euler’s totient function: The fiber above $n$ is the set of integers to which Euler’s totient function assigns the value $n$ .
Squarefree: An integer is squarefree when no prime factor is repeated in its factorization.

Keywords : Prime factors · Poisson–Dirichlet · Euler’s totient function

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that the normalized ordered logarithms of the prime factors of $p - 1$ , counted with multiplicity, converge jointly to the Poisson–Dirichlet law $PD (1)$ 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 $n^{1 - ε}$ totient preimages, for every ε > 0.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Weighted dilation graphs, smooth shifted primes and totient fibers

24/09/2026

To be verified

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 $n^{1 - ε}$ preimages. We also show that, for every fixed δ > 0, there are at least $x^{1 - o (1)}$ primes p in $2 x < p \leq 5 x$ whose predecessors have no prime factor exceeding x^δ.

PDF GitHub folder

The Poisson-Dirichlet law for prime predecessors

24/09/2026

To be verified

For a prime p chosen uniformly from $3 \leq p \leq x$ , list the prime factors of $p - 1$ in decreasing order, with multiplicity. As $x \to \infty$ , their logarithms, divided by $lo g (p - 1)$ , converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.

PDF GitHub folder

Prime Predecessors with an Even Number of Prime Factors

17/09/2026

To be verified

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 $μ (p - 1) = 1$ .

PDF GitHub folder