Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and for every positive integer . The conjecture is one of Landau's problems (1912) on prime numbers; as of 2024[update], the conjecture has neither been proved nor disproved.
Does there always exist at least one prime between and ?
If Legendre's conjecture is true, the gap between any prime p and the next largest prime would be , as expressed in big O notation.[a] It is one of a family of results and conjectures related to prime gaps, that is, to the spacing between prime numbers. Others include Bertrand's postulate, on the existence of a prime between and , Oppermann's conjecture on the existence of primes between , , and , Andrica's conjecture and Brocard's conjecture on the existence of primes between squares of consecutive primes, and Cramér's conjecture that the gaps are always much smaller, of the order . If Cramér's conjecture is true, Legendre's conjecture would follow for all sufficiently large n. Harald Cramér also proved that the Riemann hypothesis implies a weaker bound of on the size of the largest prime gaps.[1]
By the prime number theorem, the expected number of primes between and is approximately , and it is additionally known that for almost all intervals of this form the actual number of primes (OEIS: A014085) is asymptotic to this expected number.[2] Since this number is large for large , this lends credence to Legendre's conjecture.[3] It is known that the prime number theorem gives an accurate count of the primes within short intervals, either unconditionally[4] or based on the Riemann hypothesis,[5] but the lengths of the intervals for which this has been proven are longer than the intervals between consecutive squares, too long to prove Legendre's conjecture.
It follows from a result by Ingham that for all sufficiently large , there is a prime between the consecutive cubes and .[6] Dudek proved that this holds for all .[7]
Dudek also proved that for and any positive integer , there is a prime between and . Mattner lowered this to [8] which was further reduced to by Cully-Hugill.[9]
Baker, Harman, and Pintz proved that there is a prime in the interval for all large .[10]
A table of maximal prime gaps shows that the conjecture holds to at least , meaning .[11]