A Note on Large Prime Gaps

A prime gap is the difference between consecutive prime numbers. The $n^{\text{th}}$ prime gap, denoted $g_{n}$, is calculated by subtracting the $n^{\text{th}}$ prime from the $(n+1)^{\text{th}}$ prime: $g_{n}=p_{n+1}-p_{n}$. Cram{\'e}r's conjecture is a prominent unsolved problem in pure mathematics concerning prime gaps. The conjecture says that prime gaps are asymptotically bounded by $\mathrm{O}(\log^2 p_n)$. This paper presents a disproof of Cram\'{e}r's conjecture, which posits that the maximal gap $g_n$ between consecutive primes $p_n$ and $p_{n+1}$ satisfies $g_n = O(\log^2 p_n)$. By contradiction, we demonstrate that the conjecture leads to an inconsistent asymptotic regime for prime gaps. The result highlights a fundamental mismatch between the conjectured gap size and the actual distribution of primes. Our findings have significant implications for number theory, particularly in the study of large gaps between primes and related conjectures such as the Riemann Hypothesis and the Hardy-Littlewood conjectures. The disproof suggests that alternative models or stronger bounds may be necessary to accurately describe the maximal growth of prime gaps, opening new directions for future research in analytic number theory.