Legendre's conjecture

Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between $n^{2}$ and $(n+1)^{2}$ for every positive integer $n$ . ^[1] The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open problems on the spacing of prime numbers.

Unsolved problem in mathematics:

Does there always exist at least one prime between

n^{2}

and

(n+1)^{2}

?

(more unsolved problems in mathematics)

Prime gaps

If Legendre's conjecture is true, the gap between any prime p and the next largest prime would be $O({\sqrt {p}}\,)$ , 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 $n$ and $2n$ , Oppermann's conjecture on the existence of primes between $n^{2}$ , $n(n+1)$ , and $(n+1)^{2}$ , 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 $(\log p)^{2}$ . 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 $O({\sqrt {p}}\log p)$ on the size of the largest prime gaps.^[2]

Plot of the number of primes between n² and (n + 1)² OEIS: A014085

By the prime number theorem, the expected number of primes between $n^{2}$ and $(n+1)^{2}$ is approximately $n/\ln n$ , 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.^[3] Since this number is large for large $n$ , this lends credence to Legendre's conjecture.^[4] It is known that the prime number theorem gives an accurate count of the primes within short intervals, either unconditionally^[5] or based on the Riemann hypothesis,^[6] 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.

Partial results

It follows from a result by Ingham that for all sufficiently large $n$ , there is a prime between the consecutive cubes $n^{3}$ and $(n+1)^{3}$ .^[7]^[8] Dudek proved that this holds for all $n\geq e^{e^{33.3}}$ .^[9]

Dudek also proved that for $m=4.971\times 10^{9}$ and any positive integer $n$ , there is a prime between $n^{m}$ and $(n+1)^{m}$ . Mattner lowered this to $m=1438989$ ^[10] which was further reduced to $m=155$ by Cully-Hugill.^[11]

Baker, Harman, and Pintz proved that there is a prime in the interval $[x-x^{21/40},\,x]$ for all large $x$ .^[12]

A table of maximal prime gaps shows that the conjecture holds to at least $n^{2}=4\cdot 10^{18}$ , meaning $n=2\cdot 10^{9}$ .^[13]

Notes

^ This is a consequence of the fact that the difference between two consecutive squares is of the order of their square roots.

References

^ Legendre, Adriene Marie (1808). Essai sur la Théorie des Nombres (in French) (2 ed.). Paris: Chez Courcier. pp. 405–406.
^ Stewart, Ian (2013), Visions of Infinity: The Great Mathematical Problems, Basic Books, p. 164, ISBN 9780465022403.
^ Bazzanella, Danilo (2000), "Primes between consecutive squares" (PDF), Archiv der Mathematik, 75 (1): 29–34, doi:10.1007/s000130050469, MR 1764888, S2CID 16332859
^ Francis, Richard L. (February 2004), "Between consecutive squares" (PDF), Missouri Journal of Mathematical Sciences, 16 (1), University of Central Missouri, Department of Mathematics and Computer Science: 51–57, doi:10.35834/2004/1601051; see p. 52, "It appears doubtful that this super-abundance of primes can be clustered in such a way so as to avoid appearing at least once between consecutive squares."
^ Heath-Brown, D. R. (1988), "The number of primes in a short interval", Journal für die Reine und Angewandte Mathematik, 1988 (389): 22–63, doi:10.1515/crll.1988.389.22, MR 0953665, S2CID 118979018
^ Selberg, Atle (1943), "On the normal density of primes in small intervals, and the difference between consecutive primes", Archiv for Mathematik og Naturvidenskab, 47 (6): 87–105, MR 0012624
^ OEIS: A060199
^ Ingham, A. E. (1937). "On The Difference Between Consecutive Primes". The Quarterly Journal of Mathematics. os-8 (1): 255–266. doi:10.1093/qmath/os-8.1.255. ISSN 0033-5606.
^ Dudek, Adrian (December 2016), "An explicit result for primes between cubes", Funct. Approx., 55 (2): 177–197, arXiv:1401.4233, doi:10.7169/facm/2016.55.2.3, S2CID 119143089
^ Mattner, Caitlin (2017). Prime Numbers in Short Intervals (BSc thesis). Australian National University. doi:10.25911/5d9efba535a3e.
^ Cully-Hugill, Michaela (2023-06-01). "Primes between consecutive powers". Journal of Number Theory. 247: 100–117. arXiv:2107.14468. doi:10.1016/j.jnt.2022.12.002. ISSN 0022-314X.
^ Baker, R. C.; Harman, G.; Pintz, J. (2001), "The difference between consecutive primes, II" (PDF), Proceedings of the London Mathematical Society, 83 (3): 532–562, doi:10.1112/plms/83.3.532, S2CID 8964027
^ Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2014), "Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$ " (PDF), Mathematics of Computation, 83 (288): 2033–2060, doi:10.1090/S0025-5718-2013-02787-1, MR 3194140.