Cramer's Prime Gap Conjecture

Establish that the gap between consecutive prime numbers satisfies p_{n+1} − p_n = O((log p_n)^2) for all n, as asserted by Cramer's conjecture.

Background

In developing tools for rare-case hardness, the paper requires many primes of similar size to balance correctness across fields. The prime number theorem alone is insufficient, and Cramer's conjecture would provide a strong bound on gaps between consecutive primes, ensuring a dense set of primes in short intervals. Since this conjecture remains open, the paper instead relies on the best known unconditional bound on prime gaps to proceed.

References

The conjecture of states that $p_{n+1} - p_n = O \left( (\log p_n)2 \right)$\footnote{$p_n$ refers to the $n$th prime number.} and this would suffice for us. However, this problem is open and is stronger than the upper bounds implied by the Riemann hypothesis \citep{Riemann1859}.

— New Techniques for Constructing Rare-Case Hard Functions  (2411.09597 - Nareddy et al., 2024) in Section 2.5 (The Distribution of Primes)

To refine (\ref{eq: asymptotic bound}), we need an upper bound on $d(s)$. To that end, we will assume the following well-known open conjecture in number theory.

— Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds  (2608.25889 - Fushida-Hardy et al., 26 Aug 2026) in Conjecture 4.2, Section 4