Andrica's Conjecture

Let \(p_n\) denote the \(n\)-th prime. Then \(\sqrt{p_{n+1}} - \sqrt{p_n} < 1\) for every \(n\): successive square roots of primes never drift a full unit apart.

Proposed by the Romanian mathematician Dorin Andrica in 1986. It is the strongest of the classical square-root-scale gap conjectures: Andrica's inequality implies Legendre's conjecture, since a gap \(g_n = p_{n+1} - p_n\) satisfying \(g_n < 2\sqrt{p_n} + 1\) always lands a prime between consecutive squares. Like Legendre's, it has been verified computationally to vast bounds (see the prime-gap records) while remaining theoretically untouched — even the Riemann hypothesis implies it only with room to spare, and no unconditional approach is in sight.