Legendre's conjecture

Legendre's conjecture states that for every positive integer n, at least one prime lies strictly between n² and (n+1)². It remains unproved.

Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The conjecture asks whether every interval between consecutive squares contains one.
Bertrand's postulate: The theorem that every integer greater than 1 has a prime strictly between it and twice itself. It guarantees primes in shorter multiplicative ranges, but not specifically between consecutive squares.
Adrien-Marie Legendre: A French mathematician whose work included number theory, analysis, and the method of least squares. The conjecture is named after Legendre, who stated it in work on prime numbers.
Mills' theorem: A theorem asserting the existence of a constant whose successive powers produce primes by taking integer parts. It demonstrates a proved prime-existence pattern, unlike Legendre's elementary-looking interval claim.
Consecutive integers: Integers that follow one another with no integer between them. The conjecture compares squares formed from consecutive positive integers.
Prime number theorem: A theorem stating that the number of primes at most x is asymptotic to x divided by its natural logarithm. Its estimate predicts many primes near large squares but does not prove one in every such interval.
Essai sur la théorie des nombres: Legendre's 1798 treatise on number theory, expanded in a second edition in 1808. Legendre discussed the prime-distribution claim in this work.
Prime-rich intervals: Intervals whose lengths or locations ensure that they contain one or more primes. The unresolved issue is whether every consecutive-square interval is prime-rich.
Integer interval: A set of integers bounded by specified endpoints, with endpoint inclusion determined by convention. Legendre's claim concerns primes strictly inside each pair of consecutive squares.
Cramér's conjecture: A conjecture that the largest prime gap near x is asymptotically at most (log x)². Such a gap bound would eventually be smaller than the distance between consecutive squares.

Linked from 5 pages