OpenAI research collection · 1 manuscript · Source checked 2026-10-19
The idea in plain language
Prime numbers sometimes appear close together and sometimes far apart. This paper claims that gaps exceeding any fixed multiple of the typical logarithmic scale occur with a positive frequency, once enough primes are considered.
Why it matters
It sharpens the picture of how unevenly primes are distributed. The frequency depends on the chosen multiple and is not a single universal percentage.
Read the technical source claim
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed Clog 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.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
Positive lower density of large prime gaps
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed Clog 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…
Full source abstract
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed Clog 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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.