Legendre's conjecture

If Legendre's conjecture is true, the gap between any prime p and the next largest prime would be $O(\backslash sqrt\{p\}\backslash ,)$, as expressed in big O notation.{{efn|This is a consequence of the fact that the difference between two consecutive squares is of the order of their square roots.}} 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 $(\backslash log\; p)^2$. 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 $O(\backslash sqrt\; p\backslash log\; p)$ on the size of the largest prime gaps.{{citation|title=Visions of Infinity: The Great Mathematical Problems|first=Ian|last=Stewart|authorlink=Ian Stewart (mathematician)|publisher=Basic Books|year=2013|isbn=9780465022403|page=164|url=https://books.google.com/books?id=dzdSy3diraUC&pg=PA164}}.

File:Plot of number of primes between consecutive squares.png

By the prime number theorem, the expected number of primes between $n^2$ and $(n+1)^2$ is approximately $n/\backslash ln\; n$, and it is additionally known that for almost all intervals of this form the actual number of primes ({{OEIS2C|id=A014085}}) is asymptotic to this expected number.{{citation

| last = Bazzanella | first = Danilo

| doi = 10.1007/s000130050469

| issue = 1

| journal = Archiv der Mathematik

| mr = 1764888

| pages = 29–34

| title = Primes between consecutive squares

| volume = 75

| year = 2000| s2cid = 16332859

| url = http://porto.polito.it/1397858/2/Primes_between_squares.pdf

}} Since this number is large for large $n$, this lends credence to Legendre's conjecture.{{citation

| last = Francis | first = Richard L.

| date = February 2004

| doi = 10.35834/2004/1601051

| issue = 1

| journal = Missouri Journal of Mathematical Sciences

| pages = 51–57

| publisher = University of Central Missouri, Department of Mathematics and Computer Science

| title = Between consecutive squares

| volume = 16| doi-access = free

| url = https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-16/issue-1/Between-Consecutive-Squares/10.35834/2004/1601051.pdf

}}; 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." It is known that the prime number theorem gives an accurate count of the primes within short intervals, either unconditionally{{citation

| last = Heath-Brown | first = D. R.

| doi = 10.1515/crll.1988.389.22

| journal = Journal für die Reine und Angewandte Mathematik

| mr = 953665

| pages = 22–63

| title = The number of primes in a short interval

| volume = 1988

| year = 1988| issue = 389

| s2cid = 118979018

| url = https://hal.archives-ouvertes.fr/hal-01108692/file/16Article2.pdf

}} or based on the Riemann hypothesis,{{citation

| last = Selberg | first = Atle

| issue = 6

| journal = Archiv for Mathematik og Naturvidenskab

| mr = 12624

| pages = 87–105

| title = On the normal density of primes in small intervals, and the difference between consecutive primes

| volume = 47

| year = 1943}} 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 $n$, there is a prime between the consecutive cubes $n^3$ and $(n+1)^3$.{{OEIS2C|id=A060199}}{{cite journal | last=Ingham | first=A. E. | author-link=Albert Ingham | title=On The Difference Between Consecutive Primes | journal=The Quarterly Journal of Mathematics | volume=os-8 | issue=1 | date=1937 | issn=0033-5606 | doi=10.1093/qmath/os-8.1.255 | pages=255–266| bibcode=1937QJMat...8..255I }} Dudek proved that this holds for all $n\; \backslash ge\; e^\{e^\{33.3\}\}$.{{Citation |last=Dudek |first=Adrian |title=An explicit result for primes between cubes |date=December 2016 |journal=Funct. Approx. |volume=55 |issue=2 |pages=177–197 |arxiv=1401.4233 |doi=10.7169/facm/2016.55.2.3 |s2cid=119143089}}

Dudek also proved that for $m\; =\; 4.971\backslash times10^9$ and any positive integer $n$, there is a prime between $n^m$ and $(n+1)^m$. Mattner lowered this to $m\; =\; 1438989$ {{cite thesis |last=Mattner |first=Caitlin |date=2017 |title=Prime Numbers in Short Intervals |url=https://openresearch-repository.anu.edu.au|degree=BSc |publisher=Australian National University |doi=10.25911/5d9efba535a3e|language=en }} which was further reduced to $m\; =\; 155$ by Cully-Hugill.{{Cite journal |last=Cully-Hugill |first=Michaela |date=2023-06-01 |title=Primes between consecutive powers |url=https://www.sciencedirect.com/science/article/pii/S0022314X23000112 |journal=Journal of Number Theory |volume=247 |pages=100–117 |doi=10.1016/j.jnt.2022.12.002 |issn=0022-314X|arxiv=2107.14468 }}

Baker, Harman, and Pintz proved that there is a prime in the interval $[x-x^\{21/40\},\backslash ,x]$ for all large $x$.{{citation |last1=Baker |first1=R. C. |first2=G. |last2=Harman |first3=J. |last3=Pintz |year=2001 |title=The difference between consecutive primes, II |journal=Proceedings of the London Mathematical Society |volume=83 |issue=3 |pages=532–562 |doi=10.1112/plms/83.3.532 |s2cid=8964027 |url=http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf }}

A table of maximal prime gaps shows that the conjecture holds to at least $n^2=4\backslash cdot10^\{18\}$, meaning $n=2\backslash cdot10^9$.{{citation

| last1 = Oliveira e Silva | first1 = Tomás

| last2 = Herzog | first2 = Siegfried

| last3 = Pardi | first3 = Silvio

| doi = 10.1090/S0025-5718-2013-02787-1

| issue = 288

| journal = Mathematics of Computation

| mr = 3194140

| pages = 2033–2060

| title = Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\backslash cdot10^\{18\}$

| volume = 83

| year = 2014| doi-access = free

| url = https://www.openaccessrepository.it/record/100345/files/fulltext.pdf

}}.

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• {{mathworld |urlname=LegendresConjecture |title=Legendre's conjecture|mode=cs2}}

{{Prime number conjectures}}

Category:Conjectures about prime numbers

Category:Squares in number theory

Category:Unsolved problems in number theory

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