Prime Gaps
The irregular silences between consecutive primes — random-looking, yet constrained by deep arithmetic.
Prime gaps — the differences pn+1 − pn between consecutive primes — appear almost random, yet are governed by profound constraints. All gaps after the first must be even (since every prime >2 is odd). Twin primes (gap = 2) appear to continue forever, but this remains unproven: one of mathematics’ most famous open conjectures.
The scatter plot shows each gap vs. the prime it follows. Notice how the upper envelope grows roughly as ln(p), matching the prime number theorem’s prediction that primes near N are spaced about ln(N) apart on average. The Ulam spiral arranges integers in a square spiral and marks primes — revealing unexpected diagonal patterns that no one fully understands.