Prime-factor statistics of $p-1$

The prime factors of p − 1, for a random prime p, are claimed to split like a randomly broken stick.

3 papers September 17, 2026 Explained by AI

Pick a prime p at random and factor p − 1. Measure each prime factor by its share of the logarithm of p − 1. The second paper claims these shares follow the Poisson–Dirichlet law, the pattern of a stick broken at random points, as Ford, Konyagin and Luca conjectured. The first paper claims to prove Erdős's conjecture that some n are values of Euler's totient in nearly n ways. A third paper adds a parity result.

First step

1956

P. Erdős, who conjectured that some values of Euler's totient function are taken nearly n times

Significance

Major

It claims two named conjectures on shifted primes, one of Erdős and one of Ford, Konyagin and Luca.

Trust

Claimed, not checked

Field

Number theory

3 papers

Best proven exponent c with totient fibers larger than n^c infinitely often

Year	Who	Value
1998	Baker and Harman	0.7039
2022	Lichtman	0.7156
2026	This paper	1

Higher is better. 1 is the largest possible exponent.

The explanation

Take the prime 31. Then 31 − 1 = 30, which factors as 5 × 3 × 2. Measure each prime factor by its logarithm, divided by the logarithm of 30. The three shares come out near 0.473, 0.323 and 0.204. They add up to exactly 1, because the logarithms of the factors add up to the logarithm of 30. So the prime factors of p − 1 split a unit length into pieces.

For ordinary whole numbers chosen at random, the shape of this split has long been known. Dickman described the largest piece in 1930, and Billingsley found the joint law of the large pieces in 1972Source 1. The answer is the Poisson–Dirichlet law (the paper calls it PD(1)). Here is a picture of it. Break a stick of length 1 at a uniformly random point and set one part aside. Break the rest the same way, again and again. Then sort the pieces from largest to smallest.

Primes are harder. Asking that a large number divides p − 1 puts p in a fixed pattern of remainders. Standard tools count primes in such patterns only when the number is below a fixed power of xSource 2. But the full split involves products of factors right up to x. Ford, Konyagin and Luca conjectured in 2010 that p − 1 still follows PD(1)Source 3. Before this, Bharadwaj and Rodgers had shown it only under an unproved hypothesis about primes in patternsSource 4.

The second paper claims that conjecture in fullSource 3. For a prime p chosen evenly up to x, the sorted shares converge to PD(1) as x growsSource 5. That covers the largest share, the first two together, and any fixed number of them. One consequence concerns smooth predecessors, where every prime factor of p − 1 is small. It confirms Granville’s predicted share of primes p up to x with every prime factor of p − 1 at most x^(1/u)Source 6.

The first paper comes first in the chain of proofs. Fix any δ between 0 and 1/4. The paper claims at least x^(1 − o(1)) primes between 2x and 5x whose p − 1 has no prime factor above x^δSource 7. Here x^(1 − o(1)) means nearly x, up to a factor that grows slower than any power. Earlier work reached only smoothness thresholds near x^0.2961 and x^0.2844Source 8.

That input bears on Euler’s totient φ(m). It counts the numbers from 1 to m sharing no factor with m. Several m can give the same value. For example, φ(m) = 24 for exactly ten values of m. They are 35, 39, 45, 52, 56, 70, 72, 78, 84 and 90. Fix any ε > 0. The paper claims infinitely many n are hit by more than n^(1−ε) values of mSource 9. If correct, this settles a conjecture of Erdős, recorded by Pomerance in 1980Source 10. The exponent is the best possible, since the count can never exceed about nSource 10. Earlier proven exponents were 0.7039 and then 0.7156Source 11 Source 12.

A third paper claims infinitely many primes p of a special kindSource 13. For them, p − 1 has no repeated prime factor and an even number of prime factors. Parity is a classic blind spot of sieve methods (ways of counting by removing multiples of small primes)Source 14. It builds on the first paper’s toolsSource 15.

The main new step in the second paper is a counting estimate for products of two partsSource 16. The difficulty is that one side must carry marks recording chosen small prime factors of p − 1. The paper claims this works while the other side is only roughly controlled. The result is a limit law: it does not give how fast the limit is reached. It also does not prove the extra independence assumptions behind Ford, Konyagin and Luca’s prime-chain modelSource 17.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. The three papers depend on each other. No specialist review has been published.

What it doesn't show. This does not give rates of convergence, and it does not prove the independence assumptions used in models of prime chains.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major only if the chain of three linked papers holds. No Lean formalization is listed and no specialist review is published. The main law gives a limit without a rate.

Words used

Predecessor: For a prime p, the number p − 1 just before it.
Poisson–Dirichlet law: The random way a stick of length 1 splits when you break it at random points again and again, with pieces sorted by size.
Smooth number: A number whose prime factors are all below a chosen limit.
Euler’s totient: φ(m), the count of numbers from 1 to m that share no factor with m.
Totient fiber: All the numbers m that give the same totient value n.
Sieve method: A way of counting numbers with a property by removing multiples of small primes step by step.

The original papers (3)

Weighted dilation graphs, smooth shifted primes and totient fibers (PDF, opens in a new tab)
September 24, 2026
The Poisson-Dirichlet law for prime predecessors (PDF, opens in a new tab)
September 24, 2026
Prime Predecessors with an Even Number of Prime Factors (PDF, opens in a new tab)
September 17, 2026

All quoted lines (17)

Billingsley proved the joint limiting law of the ordered large prime divisors.
The Poisson-Dirichlet law for prime predecessors Back to text
Standard distribution estimates give useful information for products below a fixed power of $x$ , whereas the full factor partition involves products throughout the range below $x$ .
The Poisson-Dirichlet law for prime predecessors Back to text
Theorem … resolves positively the conjecture of Ford, Konyagin and Luca.
The Poisson-Dirichlet law for prime predecessors Back to text
For shifted primes this gives the full law under the Elliott–Halberstam conjecture, as anticipated in.
The Poisson-Dirichlet law for prime predecessors Back to text
The limit holds through all real $x$ and uses ordinary equal weighting of the primes.
The Poisson-Dirichlet law for prime predecessors Back to text
Theorem … resolves this fixed- $u$ conjecture positively and proves the complete joint limiting law.
The Poisson-Dirichlet law for prime predecessors Back to text
For every fixed $0<\delta<1/4$ , as $x\to\infty$ , $\#\{p:\ p\text{ prime},\ 2x<p\le5x, \ P^+(p-1)\le x^\delta\} \ge x^{1-o(1)}.$
Weighted dilation graphs, smooth shifted primes and totient fibers Back to text
Baker and Harman and Lichtman obtained lower bounds of the form $x/(\log x)^C$ for primes with smooth predecessors, at thresholds $x^{0.2961}$ for $p\le x$ and $x^{0.2844}$ for $x<p\le2x$ , respectively.
The Poisson-Dirichlet law for prime predecessors Back to text
For every real $\eps>0$ , there are infinitely many positive integers $n$ such that $g(n)>n^{1-\eps}.$
Weighted dilation graphs, smooth shifted primes and totient fibers Back to text
An elementary upper bound is $g(n)\ll_\eta n^{1+\eta}$ for every $\eta>0$ ; we recall it in …. Thus the power exponent of $n$ in the theorem is optimal.
Weighted dilation graphs, smooth shifted primes and totient fibers Back to text
Baker and Harman obtained exponent $0.2961$ for shifted primes, with the corresponding totient multiplicity exponent $0.7039$ .
Weighted dilation graphs, smooth shifted primes and totient fibers Back to text
Lichtman proved a lower bound $x/(\log x)^C$ for $x<p\le2x$ and $P^+(p-1)\le x^\beta$ whenever $\beta>\frac{15}{32\sqrt e}=0.284311\ldots,$ and obtained multiplicity exponent $0.7156$ .
Weighted dilation graphs, smooth shifted primes and totient fibers Back to text
There are infinitely many primes $p$ such that $p-1$ is squarefree and $\Omega(p-1)$ is even.
Prime Predecessors with an Even Number of Prime Factors Back to text
Prime-factor parity is a classical obstruction in sieve theory: congruence information alone need not distinguish the two Liouville signs.
Prime Predecessors with an Even Number of Prime Factors Back to text
Our structural inputs are the dilation-graph transference, ideal-kernel, weighted-sieve, and proxy estimates of the companion paper Weighted dilation graphs, smooth shifted primes and totient fibers, hereafter S. Their hypotheses and the conclusions needed here are stated where they enter the proof.
Prime Predecessors with an Even Number of Prime Factors Back to text
The main new estimate is Theorem …: a marked Type II estimate in which only the predecessor side carries divisor marks, namely weights recording selected small prime divisors.
The Poisson-Dirichlet law for prime predecessors Back to text
Theorem … supplies the factorization law at the root of their probabilistic model; the recursive independence assumptions used for later generations are additional hypotheses.
The Poisson-Dirichlet law for prime predecessors Back to text

Other sources

013Ostmann’s inverse Goldbach conjectureNumber theory
010Unrestricted pro-modularity at the prime twoNumber theory
012Independent largest prime factors of consecutive integersNumber theory
021A quadratic bound for Jacobsthal’s functionNumber theory
022The weak inhomogeneous Duffin–Schaeffer conjectureNumber theory

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