Positive lower density of large prime gaps at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Positive lower density of large prime gaps

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	primes, fractions, patience	Levels:	1
Category:	Number theory	Lean version:	YES! ✔
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Positive lower density of large prime gaps. For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C\log p_n$, 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.

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LEVEL 1 Positive lower density of large prime gaps

released 2026-09-25 | 1 theorem · 5 lemmas · 11 proofs · 7,213 words | PLAY LEVEL 1 » (pdf)

For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $C\log p$, 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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