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Cramér’s Conjecture: Limsup of Normalized Prime Gaps Equals One

Determine whether limsup_{n→∞} (p_{n+1} − p_n) / (log^2 p_n) equals 1, where p_n denotes the nth prime number and log denotes the natural logarithm.

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Background

Cramér’s conjecture, arising from a probabilistic model, predicts the maximal order of prime gaps to be about log2 p; establishing the limsup equal to 1 is a central open problem in prime distribution.

In the paper, this conjecture is cited alongside other major conjectures to contextualize the paper of upper bounds on prime gaps; the author’s claimed bound relates to controlling such normalized gaps.

References

Cram{e}r(1936) conjectured that given a prime $p_n$, then $\limsup_{n \to \infty} \frac{p_{n+1}-p_n}{\log2{p_n}=1$(); on the other hand, Firoozbakht(1982) conjectured that if $p_n$ is the $n$th prime, then $p_{n+1}n < p_n{n+1}$, both of which imply a prime gap stronger than the previous ones.

— On the Maximal Gap between Primes (2510.17065 - Wang, 20 Oct 2025) in Section 1 (Introduction)