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Math-2338 · mathematics · Number theory

Exponential Distribution of Normalized Prime Gaps

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
If $p_n$ is the $n$th prime, then the normalized prime gaps $(p_{n+1}-p_n)/\log p_n$ should be asymptotically distributed like an exponential random variable with density $e^{-x}$ for $0\leq x<\infty$.

Formulation references: [S1], [S2].

Short English statement

If $p_n$ is the $n$th prime, then the normalized prime gaps $(p_{n+1}-p_n)/\log p_n$ should be asymptotically distributed like an exponential random variable with density $e^{-x}$ for $0\leq x<\infty$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2338-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2338

A positive lower proportion of gaps above every fixed logarithmic threshold proves nonzero tails. It neither gives the exact exponential law nor determines the limiting proportions for all normalized intervals.

Exponential Distribution of Normalized Prime Gaps

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 026; 2026-09-25), the asserted result is that each fixed threshold $C\log p$ is exceeded by a proportion of consecutive prime gaps bounded away from zero, which also gives a positive lower proportion of increases in $p_n/n$.

Mathematical statement of the claimed result

Manuscript 1.
Writing $p_n$ for the $n$th prime, the claimed positive lower density is \[ \forall C>0\ \exists c(C)>0,N_0(C)\ \forall N\ge N_0(C): \quad|\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}|\ge c(C)N. \] The density constant and threshold may depend on the fixed gap multiplier, which does not grow with $N$.

Outline of the argument

For manuscript 1 (family 026; 2026-09-25), a square of signed divisor sums gives appreciable weight to a prime in one short interval and little weight to a prime in the adjacent interval. Two second-moment bounds convert weighted mass to ordinary starts; bounded multiplicity then counts distinct consecutive gaps.

Formal verification

For manuscript 1 (family 026; 2026-09-25), the documented Lean scope is positive lower density of increases in the sequence $p_n/n$; the full large-gap theorem is not part of that selected result.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are PrimeGaps.lean.

Their named declarations include OAI.Problem344.large_gaps_and_ratio_density.

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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