Editing Prime number (section)

=== Open questions ===
{{Further|:Category:Conjectures about prime numbers}}

Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of [[Landau's problems]] from 1912 are still unsolved.<ref>{{harvnb|Guy|2013}}, [https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7 p. vii].</ref> One of them is [[Goldbach's conjecture]], which asserts that every even integer {{tmath|n}} greater than {{tmath|2}} can be written as a sum of two primes.<ref>{{harvnb|Guy|2013}}, [https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PA105 C1 Goldbach's conjecture, pp. 105–107].</ref> {{As of|2014}}, this conjecture has been verified for all numbers up to <math>n=4\cdot 10^{18}.</math><ref>{{cite journal | 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 <math>4\cdot10^{18}</math> | volume = 83 | year = 2014| doi-access = free }}</ref> Weaker statements than this have been proven; for example, [[Vinogradov's theorem]] says that every sufficiently large odd integer can be written as a sum of three primes.<ref>{{harvnb|Tao|2009}}, [https://books.google.com/books?id=NxnVAwAAQBAJ&pg=PA239 3.1 Structure and randomness in the prime numbers, pp. 239–247]. See especially p.&nbsp;239.</ref> [[Chen's theorem]] says that every sufficiently large even number can be expressed as the sum of a prime and a [[semiprime]] (the product of two primes).<ref>{{harvnb|Guy|2013}}, p. 159.</ref> Also, any even integer greater than 10 can be written as the sum of six primes.<ref>{{cite journal | last = Ramaré | first = Olivier | issue = 4 | journal = Annali della Scuola Normale Superiore di Pisa | mr = 1375315 | pages = 645–706 | title = On Šnirel'man's constant | url = https://www.numdam.org/item?id=ASNSP_1995_4_22_4_645_0 | volume = 22 | year = 1995 | access-date = 2018-01-23 | archive-date = 2022-02-09 | archive-url = https://web.archive.org/web/20220209175544/http://www.numdam.org/item/?id=ASNSP_1995_4_22_4_645_0 | url-status = dead }}</ref> The branch of number theory studying such questions is called [[additive number theory]].<ref>{{cite book | last = Rassias | first = Michael Th. | doi = 10.1007/978-3-319-57914-6 | isbn = 978-3-319-57912-2 | location = Cham | mr = 3674356 | page = vii | publisher = Springer | title = Goldbach's Problem: Selected Topics | url = https://books.google.com/books?id=ibwpDwAAQBAJ&pg=PP6 | year = 2017}}</ref>

Another type of problem concerns [[prime gap]]s, the differences between consecutive primes.
The existence of arbitrarily large prime gaps can be seen by noting that the sequence <math>n!+2,n!+3,\dots,n!+n</math> consists of <math>n-1</math> composite numbers, for any natural number <math>n.</math><ref>{{harvnb|Koshy|2002}}, [https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA109 Theorem 2.14, p. 109]. {{harvnb|Riesel|1994}} gives a similar argument using the [[primorial]] in place of the factorial.</ref> However, large prime gaps occur much earlier than this argument shows.<ref name="riesel-gaps"/> For example, the first prime gap of length 8 is between the primes 89 and 97,<ref>{{cite OEIS|A100964|name=Smallest prime number that begins a prime gap of at least 2n}}</ref> much smaller than <math>8!=40320.</math> It is conjectured that there are infinitely many [[twin prime]]s, pairs of primes with difference 2; this is the [[twin prime conjecture]]. [[Polignac's conjecture]] states more generally that for every positive integer <math>k,</math> there are infinitely many pairs of consecutive primes that differ by <math>2k.</math><ref name="rib-gaps">{{harvnb|Ribenboim|2004}}, Gaps between primes, pp. 186–192.</ref>
[[Andrica's conjecture]],<ref name="rib-gaps"/> [[Brocard's conjecture]],<ref name="rib-183">{{harvnb|Ribenboim|2004}}, p. 183.</ref> [[Legendre's conjecture]],<ref name="chan">{{cite journal | last = Chan | first = Joel | title = Prime time! | journal = Math Horizons | volume = 3 | issue = 3 | date = February 1996 | pages = 23–25 | jstor = 25678057| doi = 10.1080/10724117.1996.11974965 }} Note that Chan lists Legendre's conjecture as "Sierpinski's Postulate".</ref> and [[Oppermann's conjecture]]<ref name="rib-183"/> all suggest that the largest gaps between primes from 1 to {{tmath|n}} should be at most approximately <math>\sqrt{n},</math> a result that is known to follow from the Riemann hypothesis, while the much stronger [[Cramér conjecture]] sets the largest gap size at {{tmath| O((\log n)^2) }}.<ref name="rib-gaps"/> Prime gaps can be generalized to [[Prime k-tuple|prime {{tmath|k}}-tuples]], patterns in the differences among more than two prime numbers. Their infinitude and density are the subject of the [[first Hardy–Littlewood conjecture]], which can be motivated by the [[heuristic]] that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.<ref>{{harvnb|Ribenboim|2004}}, Prime {{tmath|k}}-tuples conjecture, pp. 201–202.</ref>