Prime Gap Distribution
Gaps between consecutive primes — Cramér's conjecture predicts max gap ~ (ln p)²
About: A prime gap is the difference g = p_{n+1} − p_n between consecutive primes. The prime number theorem implies average gap near p is ln(p). Twin primes (gap = 2) appear infinitely often — conjectured but unproved (Green-Tao 2004 proves arithmetic progressions; Zhang 2013 proved bounded gaps ≤ 70,000,000, improved to 246 by Maynard-Tao). Cramér's conjecture (1936) predicts the maximal gap up to N grows as (ln N)². The Hardy-Littlewood conjecture gives precise predictions for the density of each gap size. The histogram reveals an even-gap dominance (all gaps > 2 must be even) and the characteristic shape of the distribution.