Cramér's Conjecture

Claim: writing \(g_{p}\) = \(p_{next}\) − p for consecutive primes, \(g_{p}\) = O((log p)²). Cramér derived this from a random-model heuristic placing primes independently with probability 1/log n.

Unconditionally the record is \(g_{p}\) = O(\(p^{0.525}\)) (Baker–Harman–Pintz lineage); assuming RH alone one gets O(√p log p) — nowhere near polylogarithmic. Granville's refinement of the model suggests Cramér's constant might even fail slightly (gaps up to \(2e^{−γ}\)(log p)² ≈ 1.12× larger), making the problem doubly instructive: the heuristic itself may need correction. Related bounded-gap triumphs (Zhang–Maynard–Polymath: infinitely often ≤ 246) show primes DO cluster — the question is how far they sometimes stray.