Legendre's conjecture

Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n² and (n + 1)² for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers; as of 2016^[update], the conjecture has neither been proved nor disproved.

Legendre's conjecture is one of a family of results and conjectures related to prime gaps, that is, to the spacing between prime numbers.

The prime number theorem implies that the actual number of primes between n² and (n + 1)²  is asymptotic to n/ln(n). Since this number is large for large n, this lends credence to Legendre's conjecture.

If Legendre's conjecture is true, the gap between any prime p and the next largest prime would always be at most on the order of ${\displaystyle {\sqrt {p}}}$; in big O notation, the gaps are ${\displaystyle O({\sqrt {p}})}$. Two stronger conjectures, Andrica's conjecture and Oppermann's conjecture, also both imply that the gaps have the same magnitude. It does not, however, provide a solution to the Riemann Hypothesis, but rather strengthens one of the implications of its correctness.

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