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Math-2786 · mathematics · Number theory

The joint Dickman law for consecutive integers

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For fixed a,b in (0,1), is the natural density of integers $n>=2$ with largest prime factor of n at most $n^a$ and that of n+1 at most $n^b$ equal to rho(1/a)rho(1/b)? Here $\rho (u)=1$ on [0,1] and u $\rho '(u)=-\rho (u-1)$ for $u>1$.

Short English statement

For fixed a,b in (0,1), is the natural density of integers $n>=2$ with largest prime factor of n at most $n^a$ and that of n+1 at most $n^b$ equal to rho(1/a)rho(1/b)? Here $\rho (u)=1$ on [0,1] and u $\rho '(u)=-\rho (u-1)$ for $u>1$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2786-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2786

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 012; 2026-09-24).

The joint Dickman law for consecutive integers

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 012; 2026-09-24), the asserted result is that consecutive integers have independent limiting largest-prime-factor statistics with Dickman marginals, and either ordering of their largest factors has density one half.

Mathematical statement of the claimed result

Manuscript 1.
Let $P^+(n)$ be the largest prime factor and $\rho$ the Dickman function, equal to one on $[0,1]$, zero on negative arguments and satisfying $u\rho'(u)+\rho(u-1)=0$ for $u>1$. For every fixed $a,b\in(0,1)$, the claimed joint law is \[ \lim_{X\to\infty}\frac1X|\{2\le n\le X:P^+(n)\le n^a,\ P^+(n+1)\le n^b\}| =\rho(1/a)\rho(1/b). \] The limit uses ordinary unweighted counting through all real $X$, not logarithmic averaging.

Outline of the argument

For manuscript 1 (family 012; 2026-09-24), a positive divisor weight amplifies a putative mixed correlation into quadratic energy. An independent split model controls its first two moments, while residue smoothing suppresses nonconstant modes and forces the amplified energy to vanish.

Formal verification

For manuscript 1 (family 012; 2026-09-24), OpenAI lists Lean statements for this assertion: consecutive integers have independent limiting largest-prime-factor statistics with Dickman marginals, and either ordering of their largest factors has density one half.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are JointDickman.lean.

Their named declarations include OAI.JointDickmanPaper.decreasing_order, OAI.JointDickmanPaper.increasing_order, OAI.JointDickmanPaper.joint_law.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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