For every integer \(n > 1\) there is at least one prime \(p\) with \(n(n-1) < p < n^2\) and at least one prime \(q\) with \(n^2 < q < n(n+1)\). In particular there is always a prime between consecutive squares, so Oppermann's conjecture implies Legendre's.
Formulated by the Danish mathematician, actuary and politician Ludvig Oppermann in the late 19th century. It sharpens Legendre's conjecture from one prime per square-interval to one in each half: the intervals \([n(n-1), n^2]\) and \([n^2, n(n+1)]\) each contain a prime. Like its siblings it is computationally confirmed far beyond doubt but theoretically open; Cramér's probabilistic model predicts it comfortably, and any proof would have to beat the Baker–Harman–Pintz exponent substantially.
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