Positive lower density of large prime gaps

1 · The essentials

Prime numbers do not arrive at evenly spaced intervals. This result concerns how often the distance from one prime to the next is large compared with the logarithm of the smaller prime. Choose any fixed positive multiplier, however large. The manuscript shows that a positive share of the gaps exceed that multiplier times the logarithm of the smaller prime. This share stays bounded away from zero in every sufficiently long beginning of the prime list, although the bound may depend on the multiplier. A consequence answers a question of Erdős and Prachar: dividing each prime by its position in the list gives a value that increases at a positive share of positions, in the same lasting sense. The main result has a Lean formalization, though the supplied description does not specify its exact coverage.

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

As an imperfect image, imagine stops along a road, with the distance between successive stops compared against a benchmark that changes with location. For any fixed multiplier of that benchmark, the result says that unusually long stretches account for at least some positive share of every sufficiently long starting portion of the route—not merely that such stretches occasionally occur.

Why it matters

The result gives a persistent frequency statement about large prime gaps, rather than merely asserting that they occur. Its consequence for increases of $p_{n} / n$ answers the question attributed in the source to Erdős and Prachar.

2 · Going further

Let $p_{n}$ denote the $n$ th prime and let $g_{n} = p_{n + 1} - p_{n}$ be the next prime gap. The main statement is that for every fixed $C > 0$ , there are constants $δ (C) > 0$ and $N_{0} (C)$ such that, for every $N \geq N_{0} (C)$ , $\frac{1}{N} # {1 \leq n \leq N : g_{n} > C lo g p_{n}} \geq δ (C)$ . Thus the frequency is bounded below throughout all sufficiently large initial segments, not just along a selected sequence of endpoints.

The manuscript also shows that the set of indices for which $p_{n + 1} / (n + 1) > p_{n} / n$ has positive lower asymptotic density. It presents this as a consequence of the large-gap statement and as an answer to a question of Erdős and Prachar. Here, positive lower asymptotic density means that the fraction of qualifying indices among the first $N$ remains bounded away from zero as $N$ grows sufficiently large.

The supplied abstract states the result and this implication but does not describe the argument establishing them. No companion manuscript is supplied, so no further relationship or proof strategy can be reported from this source. The main result has a Lean formalization; the supplied information does not identify which additional claims, including the consequence, are covered.

Useful notions

Prime numbers and their ordering
Differences between consecutive numbers
Logarithms
Proportions in a growing list
Lower asymptotic density

What was known before / what is new

Before

The source identifies a question of Erdős and Prachar concerning the indices where $p_{n} / n$ increases. It supplies no earlier bounds, methods, or other historical context.

New

The manuscript shows that, for every fixed $C > 0$ , gaps larger than $C lo g p_{n}$ occupy a positive proportion of every sufficiently large initial segment of the prime sequence. The lower bound may depend on $C$ . It also gives the positive-lower-density consequence for increases of $p_{n} / n$ , answering the stated question.

Limits and assumptions

The multiplier $C$ must be fixed and positive, and the frequency guarantee applies only to sufficiently large initial segments. The source does not give a common positive bound valid for all multipliers, an explicit proportion, or an explicit starting threshold. The main result has a Lean formalization, but its exact scope is not specified; formal verification of the consequence cannot be inferred from the supplied description. No remaining open question is identified in the source.

Mini glossary

Consecutive prime gap: The difference $p_{n + 1} - p_{n}$ between two successive primes.
Initial segment: A starting portion of a sequence, consisting of its first $N$ entries.
Positive lower asymptotic density: A property of a set of indices whose proportion among the first $N$ indices stays bounded away from zero for all sufficiently large $N$ .
Lean formalization: A mathematical statement and its verification expressed in the Lean proof-assistant system.

Keywords : Prime gaps · Lower asymptotic density · Lean formalization

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)

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C lo g p_{n}$ , throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where $p_{n} / n$ increases have positive lower density, answering Erdős and Prachar.

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/026.md

Manuscripts in this family

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Positive lower density of large prime gaps

25/09/2026

To be verified

For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $C lo g p$ , where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which $p_{n} / n$ increases have positive lower asymptotic density, answering a question of Erdős and Prachar.

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