Prime Gap Distribution

Statistics of gn = pn+1 − pn — gaps between consecutive primes. Cramér's conjecture: max gap ~ (log p)². Explore the Ulam spiral and gap histograms.

Gap Plot Histogram Ulam Spiral Cramér

Range

Stats

Cramér's Conjecture (1936)

lim sup g_n / (log p_n)² = 1

Maximal prime gaps grow roughly as (log p)².

Green-Tao (2004): primes contain arbitrarily long arithmetic progressions.

Zhang (2013): infinitely many gaps < 70,000,000. Polymath: gap < 246.