Factoring one less than a prime: a Poisson–Dirichlet law

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Family 011 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Choose a large prime $p$, subtract one, and factor the result. How large should its largest prime factor be? How should the second-largest compare with it? Does requiring $p$ to be prime change the large-factor pattern expected for an ordinary integer?

Ford, Konyagin, and Luca conjectured that the normalized large factors of $p-1$ follow the same Poisson–Dirichlet law that appears for random integers. Family 011 claims that law in all finite joint distributions. Its companion manuscripts address two related questions: whether Euler’s totient function can take one value almost as many times as its size allows, and whether infinitely many prime predecessors are squarefree with an even number of prime factors.

These questions share analytic machinery, but one conclusion does not automatically imply the others. Large-factor size distributions do not detect the parity of the total number of factors. Likewise, producing many smooth predecessors is weaker than determining their limiting proportion among all primes.

2. Background and essential definitions

For an integer $n>1$, let $P^+(n)$ be its largest prime factor. The integer is $y$-smooth if $P^+(n)\le y$. Smoothness says every prime factor is small; it does not say there are few factors.

List the prime factors of $p-1$ in decreasing order, with multiplicity:

$$q_1(p)\ge q_2(p)\ge\cdots.$$

After the list ends, put $q_j(p)=1$. Define

$$V_j(p)=\frac{\log q_j(p)}{\log(p-1)}.$$

The entries are nonnegative and sum to one because the product of the prime factors is $p-1$. For $p=13$, the factors of twelve are $3,2,2$; the two occurrences of two are both counted. The normalized logarithms describe how the total multiplicative size is divided among the factors.

The Poisson–Dirichlet distribution with parameter one, written $\mathrm{PD}(1)$, can be constructed by randomly breaking a unit stick. Choose independent uniform random variables $U_1,U_2,\ldots$ on $(0,1)$, and set

$$B_1=1-U_1,\qquad B_j=\left(\prod_{i<j}U_i\right)(1-U_j).$$

At each step, remove a uniform fraction of the remaining stick. Sort the fragments into decreasing order $L_1\ge L_2\ge\cdots$. Their distribution is $\mathrm{PD}(1)$. The pieces are dependent, since their total is one; this is not a model of independent factor sizes.

Euler’s totient $\varphi(m)$ counts the integers from one to $m$ that are coprime to $m$. Its formula is

$$\varphi(m)=m\prod_{q\mid m}\left(1-\frac1q\right).$$

The fiber size $g(n)=\#\{m\ge1:\varphi(m)=n\}$ counts how often a particular value occurs. For example, $\varphi(5)=\varphi(8)=\varphi(10)=\varphi(12)=4$, so distinct inputs can share a totient.

Finally, $\Omega(n)$ counts prime factors with multiplicity, while $\omega(n)$ counts distinct prime factors. They agree for squarefree integers, which have no repeated prime factor. The Möbius function is zero on nonsquarefree integers and equals $(-1)^{\Omega(n)}$ on squarefree ones.

3. Development history and previous partial results

For uniformly sampled ordinary integers, Dickman’s work and de Bruijn’s smooth-number estimates describe the largest prime factor. Billingsley obtained the joint limiting law of the ordered large factors. Donnelly and Grimmett developed a size-biased interpretation leading to the stick-breaking description. These establish the random-integer model motivating the prime-predecessor conjecture.

The restriction to $p-1$ introduces a primality constraint. Requiring a product $d$ of selected factors to divide $p-1$ is equivalent to requiring $p\equiv1\pmod d$. Thus factor statistics depend on how primes are distributed among arithmetic progressions, sometimes with very large moduli.

Ford, Konyagin, and Luca’s work on prime chains and Pratt trees formulated the full distribution conjecture. A prime chain links primes by divisibility of successive predecessors. The factorization law would describe the first generation of a recursive prime-factor tree, though independence assumptions for later generations are additional questions.

For smooth shifted primes, Baker and Harman obtained the fixed exponent $0.2961$, together with a totient multiplicity exponent $0.7039$. Lichtman’s work improved the smoothness threshold to every fixed exponent above approximately $0.284311$, and obtained multiplicity exponent $0.7156$. Such results produce many predecessors with all factors below a specified power, but do not reach every positive exponent or a full limiting law.

Bharadwaj and Rodgers established Poisson–Dirichlet laws for sufficiently well-distributed sequences. Applied to shifted primes, their full conclusion follows under the Elliott–Halberstam conjecture; unconditional distribution estimates yield factor statistics only in a restricted region. M2 claims to obtain the complete law unconditionally.

The totient question goes back to Erdős, with Pomerance developing the connection to smooth predecessors. The parity question faces a different classical obstacle: sieve information about divisibility can fail to distinguish even from odd numbers of prime factors. Bilinear cancellation, central to the asymptotic-sieve method of Friedlander and Iwaniec, provides a way beyond divisibility counts alone. The manuscripts develop graph estimates for that purpose.

4. What the new manuscripts claim

M2: the full factor-size law

For every fixed positive integer $k$ and bounded continuous function $F:[0,1]^k\to\mathbb R$, M2, Theorem 1.1, claims

$$\lim_{x\to\infty}\frac1{\pi(x)-1} \sum_{\substack{3\le p\le x\\p\text{ prime}}} F(V_1(p),\ldots,V_k(p)) =\mathbb E F(L_1,\ldots,L_k).$$

Here $\pi(x)$ counts primes at most $x$, so $\pi(x)-1$ excludes the prime two. Primes receive ordinary equal weight, and the limit holds through all real cutoffs. This describes every fixed finite collection of largest normalized factors.

In particular, for every fixed $u\ge1$, the manuscript deduces

$$\#\{p\le x:P^+(p-1)\le x^{1/u}\} \sim\pi(x)\rho(u),$$

with $P^+(1)=1$. The notation $\sim$ means the ratio tends to one. The Dickman function $\rho$ is determined by

$$\rho(u)=1\quad(0\le u\le1),\qquad u\rho'(u)+\rho(u-1)=0\quad(u>1).$$

The parameter $u$ is fixed before $x$ grows.

M1: smooth predecessors and large totient fibers

M1, Theorem 1.2, asserts that for every fixed $0<\delta<1/4$,

$$\#\{p:2x<p\le5x,\ P^+(p-1)\le x^\delta\} \ge x^{1-o(1)}.$$

The vanishing quantity $o(1)$ may depend on $\delta$. The conclusion extends to every fixed positive $\delta$ by using a smaller exponent. It gives the full power exponent for the count, but alone does not imply a positive limiting proportion among primes.

M1, Theorem 1.1, also claims that for every $\varepsilon>0$, infinitely many integers $n$ satisfy

$$g(n)>n^{1-\varepsilon}.$$

The source recalls an elementary upper bound $g(n)\ll_\eta n^{1+\eta}$ for every $\eta>0$. Thus its result is optimal at the level of the power exponent, not a precise asymptotic for the largest fiber.

M3: a parity conclusion

M3, Theorem 1.1, claims infinitely many primes $p$ such that $p-1$ is squarefree and $\Omega(p-1)$ is even. Equivalently,

$$\mu(p-1)=1.$$

Squarefreeness makes the distinct-factor and multiplicity conventions coincide. The theorem is an infinitude statement, not a claim that half of all prime predecessors have this property.

5. Why the advance matters

Provisional importance: 4/5. The full predecessor law would establish a detailed probabilistic prediction for an arithmetically constrained sequence. It goes beyond the largest-factor statistic to describe how several large factors share the logarithmic size.

The smoothness and totient statements answer longstanding extremal questions, while the parity result confronts a limitation of ordinary sieve information. Their common methods would therefore connect distributional, extremal, and parity-sensitive problems.

The connection to totients is especially concrete. If $m$ is a product of distinct primes from a set $S$, then

$$\varphi(m)=\prod_{p\in S}(p-1).$$

When all these predecessors use a small supply of prime factors, many different choices of $S$ can produce the same product. Enough smooth predecessors allow this collision principle to create very large fibers. The score reflects the reach of the stated results if correct, without treating the different conclusions as automatic equivalents.

6. The main idea

Detect primes within a controlled factorization pattern

M1 constructs nonnegative weights on integers $u$ whose factors occupy carefully chosen size bands, then seeks prime values of $2u+1$. Some small prime factors are marked so their divisibility can be tracked. A longer factor is allowed to range over rough integers, which have no prime factors below a chosen cutoff.

Congruence estimates provide an initial sieve, but stronger cancellation is needed to remove composite candidates. The manuscript organizes multiplicative comparisons in dilation graphs. A vertex records an integer and lists of marked prime divisors; edges connect states through shifts involving products of shared and independent primes.

An ideal comparison operator uses independently sampled labels with probabilities proportional to reciprocals of primes. M1 claims that a small norm for this ideal operator transfers to cancellation in the arithmetic graph. Closed-walk expansions track repeated prime labels: a repeated divisibility condition should be charged only once, so the construction retains the relevant congruence information between appearances.

The resulting correlation estimates yield a Type II estimate, a cancellation bound for sums with two varying factor variables. Such bilinear information distinguishes prime candidates from composite factorizations more effectively than initial congruence counts. A weighted sieve then extracts enough primes with smooth predecessors. A product-and-pigeonhole argument converts that supply into the totient-fiber bound.

Obtain the entire partition, not just smooth examples

M2 uses M1’s general dilation-graph estimates but needs a separate extraction argument. Its new determinant graph controls a situation where only the predecessor side carries divisor marks. After applying Cauchy–Schwarz, pairs of factorizations become relations between primitive lattice vectors, vectors whose coordinates have no common factor.

Modulo a prime, these vectors determine projective lines. The manuscript averages their local distributions and retains repeated-line information in a signed memory expansion. A second estimate handles marks on both endpoints. Together these comparisons replace composite factor slots by suitable rough-number proxies and recover joint statistics of selected prime factors.

The first statistics are obtained inside the region where the sum of the selected logarithmic factor sizes is strictly less than one. That interior information is not yet the complete limiting law: probability could accumulate near the boundary or among omitted pieces.

M2 uses sequential size-biased sampling, choosing the next factor with probability proportional to its logarithmic size. The resulting densities are claimed to have full mass and to match the stick-breaking process. Control of the remaining unsampled mass then permits sorting and yields the ranked Poisson–Dirichlet law. The boundary argument is a necessary probabilistic step after the analytic estimates.

Preserve a parity-sensitive sign

M3 again uses M1’s graph machinery, but changes the weight. Writing $p=2u+1$, it chooses odd $u$ with an odd number of prime factors; adjoining the factor two then makes $p-1$ have even parity.

Two extensions are needed. A long factor slot must contain a prime, rather than an arbitrary rough integer. A Liouville sign, $\lambda(u)=(-1)^{\Omega(u)}$, must also survive the graph transformations. When the same dilation factor appears at both endpoints, its sign occurs twice and cancels because $\lambda(d)^2=1$.

The manuscript supplies the corresponding prime-polynomial and signed-correlation estimates, then applies a parity-selecting nonnegative weight in the sieve. It bounds the nonsquarefree contribution separately so that enough detected prime mass remains with squarefree predecessor. Smoothness by itself would not provide either this sign control or squarefreeness.

7. Scope, limitations, and open questions

All distribution and smoothness parameters are fixed before the prime cutoff tends to infinity. No uniform theorem for $u\to\infty$ or $\delta\to0$ with $x$ is asserted. The full law concerns every fixed finite collection of ranked factors, with the stated multiplicity and logarithmic normalization.

The predecessor law does not establish recursive independence throughout a Pratt tree. Nor does it determine the parity of the total number of prime factors, which may depend on many small factors invisible to a fixed-dimensional large-factor statistic. M3 supplies a separate infinitude result for that reason.

The totient theorem concerns infinitely many exceptionally popular values, not the behavior of every value or a typical fiber. Its power exponent is sharp in the stated sense, while finer maximal-order asymptotics are not given here. The manuscripts also do not assert effective thresholds for all their existence conclusions.

8. Sources and further reading

M1’s Theorems 1.1–1.2 state the fiber and smooth-predecessor results; its introduction explains the dilation graph, Type II, and sieve stages. M2’s Theorem 1.1 gives the full distribution, and Section 8 handles the passage from interior statistics to ranked factors. M3’s Theorem 1.1 and introductory outline isolate the parity extensions.

The primary papers in Section 3 place the claims beside earlier smoothness bounds and conditional distribution laws. Optional introductions to the Dickman function, Poisson–Dirichlet distributions, and Euler’s totient function provide additional background.

Source manuscripts

• M1. OpenAI, Weighted dilation graphs, smooth shifted primes and totient fibers (September 24, 2026).
• M2. OpenAI, The Poisson-Dirichlet law for prime predecessors (September 24, 2026).
• M3. OpenAI, Prime Predecessors with an Even Number of Prime Factors (September 17, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.