Independent largest prime factors of consecutive integers

1 · The essentials

Take two consecutive integers and find the largest prime divisor of each: a divisor that cannot itself be split into a product of two smaller integers greater than one. The manuscript studies the sizes of these two divisors relative to the integers, using logarithms. Its main result says that, as more and more integers are considered, these relative sizes behave statistically like independent values. Each follows the same distribution, called the Dickman distribution. The manuscript presents this as a resolution of the Erdős–Pomerance joint Dickman conjecture. It also deduces that the proportion of integers whose largest prime divisor is smaller than that of the next integer tends to $1/2$ . The main result has a Lean formalization, but the supplied information does not specify which related claims it covers.

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

As an imperfect picture, imagine two side-by-side records, each listing the relative size of the largest prime divisor of one of two neighboring integers. As the records accumulate, knowing the entry on the left no longer changes, in the limit, the statistical distribution of entries on the right. This describes an overall pattern, not a rule for predicting each row.

Why it matters

The result concerns how the sizes are distributed jointly, rather than just how each is distributed separately. It answers the conjecture named in the source positively and gives a concrete consequence: each of the two positions wins the comparison of largest prime divisors with limiting frequency one half.

2 · Going further

Write $P^{+} (n)$ for the largest prime factor of $n$ . The problem is to study jointly $X_{n} = lo g P^{+} (n) / lo g n$ and $Y_{n} = lo g P^{+} (n + 1) / lo g n$ . Both quantities therefore use the same denominator, $lo g n$ . The manuscript states that they are asymptotically independent, with Dickman marginal distributions, in ordinary natural density.

Natural density here means the proportion of integers satisfying a condition among those up to a growing bound, as that bound tends to infinity. Asymptotic independence means that the limiting joint distribution factors into the two marginal distributions: limiting frequencies for separate conditions on the two sizes multiply. The source presents this as a positive resolution of the Erdős–Pomerance joint Dickman conjecture and deduces that $P^{+} (n) < P^{+} (n + 1)$ has natural density $1/2$ .

The supplied abstract states the result and its consequence but describes neither the main idea of the argument nor its technical steps. No companion manuscript is supplied, so relationships between multiple arguments cannot be specified. The main result has a Lean formalization; its exact scope is not detailed in the available information.

Useful notions

Prime numbers and divisibility
Largest prime factors
Logarithms and relative sizes
Limiting frequencies and natural density
Probability distributions and independence

What was known before / what is new

Before

The source identifies the question as the Erdős–Pomerance joint Dickman conjecture. It provides neither a detailed history nor earlier partial results, and does not specify what was already established separately about the marginal distributions.

New

The announced contribution is asymptotic independence of the two logarithmic sizes, with Dickman marginal distributions, in ordinary natural density. The manuscript also deduces density $1/2$ for the inequality $P^{+} (n) < P^{+} (n + 1)$ . The main result has a Lean formalization.

Limits and assumptions

The statement concerns consecutive integers $n$ and $n + 1$ and the specific logarithmic sizes given above, not exact independence over a finite range of integers. It is a limiting result in ordinary natural density; the abstract supplies no convergence rate or error estimate. A Lean formalization of the main result is reported, but its exact coverage, including whether it covers the ordering consequence, is not specified. The supplied text identifies no further questions that remain open.

Mini glossary

Largest prime factor: The largest prime number dividing an integer, denoted by $P^{+} (n)$ .
Natural density: The limit, when it exists, of the proportion of integers satisfying a condition among the integers up to a growing bound.
Marginal distribution: The distribution of one quantity considered on its own within a pair.
Asymptotic independence: The property that the limiting distribution of a pair is the product of its two marginal distributions.
Dickman distribution: The distribution named in the manuscript as the limiting distribution of each of the two logarithmic sizes under study.

Keywords : Largest prime factor · Dickman distribution · Asymptotic independence

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)

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n + 1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^{+} (n) < P^{+} (n + 1)$ have density 1/2.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/012.md

Manuscripts in this family

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The joint Dickman law for consecutive integers

24/09/2026

To be verified

Let $P^{+} (n)$ denote the largest prime factor of n. We prove that $lo g P^{+} (n) / lo g n$ and $lo g P^{+} (n + 1) / lo g n$ are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering $P^{+} (n) < P^{+} (n + 1)$ has natural density 1/2.

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