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adalah prima untuk sembarang bilangan asli <math>n</math> dalam rumus pertama, dan sembarang bilangan eksponen dalam rumus kedua.<ref>{{cite journal |first=E.M. |last= Wright | author-link=E. M. Wright |title=A prime-representing function |journal=[[American Mathematical Monthly]] |volume=58 |issue=9 |year=1951 |pages=616–618 |jstor=2306356 |doi= 10.2307/2306356}}</ref> Sehingga <math>\lfloor {}\cdot{} \rfloor</math> mewakili [[fungsi lantai]], bilangan bulat terbesar yang kurang dari atau sama dengan bilangan yang dimaksud. Namun, hal ini justru tidak berguna untuk menghasilkan bilangan prima, karena bilangan prima harus dibangkitkan terlebih dahulu untuk menghitung nilai <math>A</math> atau <math>\mu.</math><ref name="matiyasevich"/>
===OpenPertanyaan questionsterbuka===
{{Further|:CategoryKategori:ConjecturesKonjektur abouttentang primebilangan numbersprima}}
ManyBanyak conjectureskonjektur revolvingtentang aboutbilangan primesprima havetelah been poseddiajukan. OftenSeringkali havingmemiliki anrumus elementary formulationdasar, manybanyak ofdari thesekonjektur conjecturesini havetelah withstoodbertahan proofselama forbeberapa decadesdekade: all four ofkeempat [[masalah Landau's problems]] fromdari tahun 1912 aremasih stillbelum unsolvedterpecahkan.<ref>{{harvnb|Guy|2013}}, [https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7 p. vii].</ref> OneSalah ofsatunya them isadalah [[konjektur Goldbach's conjecture]], whichyang assertsmenyatakan thatbahwa everysetiap evenbilangan bulat integergenap <math>n</math> greaterlebih thanbesar dari 2 canditulis besebagai writtenjumlah asdari a sum ofdua twobilangan primesprima.<ref>{{harvnb|Guy|2013}}, [https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PA105 C1 Goldbach's conjecture, pphal. 105–107].</ref> {{As of|2014}}, thisKonjektur conjectureini hastelah beendiverifikasi verifieduntuk forsemua allbilangan numbers up tohingga <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> WeakerPernyataan statementsyang thanlebih thislemah havedari beenini proventelah dibuktikan, for example,misalnya [[teorema Vinogradov's theorem]] saysyang thatmenyatakan everybahwa sufficientlysetiap largebilangan oddbulat integerganjil canbesar bedapat writtenditulis assebagai ajumlah sumdari oftiga threebilangan primesprima.<ref>{{harvnb|Tao|2009}}, [https://books.google.com/books?id=NxnVAwAAQBAJ&pg=PA239 3.1 Structure and randomness in the prime numbers, pphal. 239–247]. SeeLihat especiallyterutama phal. 239.</ref> [[Teorema Chen's theorem]] saysmenyatakan thatbahwa everysetiap sufficientlybilangan largegenap evenbesar numberdapat candinyatakan besebagai expressedjumlah asdari thesuatu sumbilangan ofprima a prime and adan [[semiprimesemiprima]] (theperkalian productdari ofdua twobilangan primesprima).<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}}</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>
Jenis masalah lain menyangkut [[celah prima]], perbedaan antara bilangan prima berurutan.
Another type of problem concerns [[prime gap]]s, the differences between consecutive primes.
TheAdanya existencecelah ofprima arbitrarilybesar largesecara primesembarang gapsdapat candilihat bedengan seenmemperhatikan bybahwa noting that the sequencebarisan <math>n!+2,n!+3,\dots,n!+n</math> consiststerdiri ofdari <math>n-1</math> compositebilangan numberskomposit, foruntuk anysembarang naturalbilangan numberasli <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 <math>1</math> to <math>n</math> 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 <math>O((\log n)^2).</math><ref name="rib-gaps"/> Prime gaps can be generalized to [[Prime k-tuple|prime <math>k</math>-tuples]], patterns in the differences between 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 <math>k</math>-tuples conjecture, pp. 201–202.</ref>
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