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Legendre's conjecture

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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, 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 and ?

Prime gaps

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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]

Plot of the number of primes between n2 and (n + 1)2 OEISA014085

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 (OEISA014085) 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.

Partial results

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It follows from a result by Ingham that for all sufficiently large , there is a prime between the consecutive cubes and .[6][7] Dudek proved that this holds for all .[8]

Dudek also proved that for and any positive integer , there is a prime between and . Mattner lowered this to [9] which was further reduced to by Cully-Hugill.[10]

Baker, Harman, and Pintz proved that there is a prime in the interval for all large .[11]

A table of maximal prime gaps shows that the conjecture holds to at least , meaning .[12]

Notes

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

References

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  1. ^ Stewart, Ian (2013), Visions of Infinity: The Great Mathematical Problems, Basic Books, p. 164, ISBN 9780465022403.
  2. ^ Bazzanella, Danilo (2000), "Primes between consecutive squares" (PDF), Archiv der Mathematik, 75 (1): 29–34, doi:10.1007/s000130050469, MR 1764888, S2CID 16332859
  3. ^ 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."
  4. ^ 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
  5. ^ 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
  6. ^ OEISA060199
  7. ^ 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.
  8. ^ 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
  9. ^ Mattner, Caitlin (2017). Prime Numbers in Short Intervals (BSc thesis). Australian National University. doi:10.25911/5d9efba535a3e.
  10. ^ 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.
  11. ^ 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
  12. ^ Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2014), "Empirical verification of the even Goldbach conjecture and computation of prime gaps up to " (PDF), Mathematics of Computation, 83 (288): 2033–2060, doi:10.1090/S0025-5718-2013-02787-1, MR 3194140.
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