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==Sejarah==
[[Berkas:Rhind Mathematical Papyrus.jpg|thumb|The [[Rhind Mathematical Papyrus]]|alt=The Rhind Mathematical Papyrus]] ▼
▲[[Berkas:Rhind Mathematical Papyrus.jpg|thumb| The [[ RhindPapirus MathematicalMatematika PapyrusRhind]]|alt= ThePapirus Matematika Rhind Mathematical Papyrus]]
The [[Rhind Mathematical Papyrus]], from around 1550 BC, has [[Egyptian fraction]] expansions of different forms for prime and composite numbers.<ref>Bruins, Evert Marie, review in ''Mathematical Reviews'' of {{cite journal | last = Gillings | first = R.J. | doi = 10.1007/BF01307175 | journal = Archive for History of Exact Sciences | mr = 0497458 | pages = 291–298 | title = The recto of the Rhind Mathematical Papyrus. How did the ancient Egyptian scribe prepare it? | volume = 12 | issue = 4 | year = 1974| s2cid = 121046003 }}</ref> However, the earliest surviving records of the explicit study of prime numbers come from [[Greek mathematics|ancient Greek mathematics]]. [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' (c. 300 BC) proves the [[infinitude of primes]] and the [[fundamental theorem of arithmetic]], and shows how to construct a [[perfect number]] from a [[Mersenne prime]].<ref name="stillwell-2010-p40">{{cite book|title=Mathematics and Its History|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|author-link=John Stillwell|edition=3rd|publisher=Springer|year=2010|isbn=978-1-4419-6052-8|page=40|url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40}}</ref> Another Greek invention, the [[Sieve of Eratosthenes]], is still used to construct lists of primes.<ref name="pomerance-sciam">{{cite journal|title=The Search for Prime Numbers|first=Carl|last=Pomerance|author-link=Carl Pomerance|journal=[[Scientific American]]|volume=247|issue=6|date=December 1982|pages=136–147|jstor=24966751|doi=10.1038/scientificamerican1282-136|bibcode=1982SciAm.247f.136P}}</ref><ref name="mollin">{{cite journal | last = Mollin | first = Richard A. | doi = 10.2307/3219180 | issue = 1 | journal = Mathematics Magazine | mr = 2107288 | pages = 18–29 | title = A brief history of factoring and primality testing B. C. (before computers) | volume = 75 | year = 2002| jstor = 3219180 }}</ref> ▼
▲The [[ RhindPapirus MathematicalMatematika PapyrusRhind]] , fromdari sekitar aroundtahun 1550 BCSM, hasmemiliki [[ Egyptianpecahan fractionMesir]] expansionsekspansi ofbentuk differentyang formsberbeda foruntuk primebilangan andprima compositedan numberskomposit.<ref>Bruins, Evert Marie, review in ''Mathematical Reviews'' of {{cite journal | last = Gillings | first = R.J. | doi = 10.1007/BF01307175 | journal = Archive for History of Exact Sciences | mr = 0497458 | pages = 291–298 | title = The recto of the Rhind Mathematical Papyrus. How did the ancient Egyptian scribe prepare it? | volume = 12 | issue = 4 | year = 1974| s2cid = 121046003 }}</ref> HoweverNamun, thecatatan earliestpertama survivingkali recordsyang ofbertahan thestudi expliciteksplisit studybilangan ofprima primeberasal numbers come fromdari [[ Greekmatematika mathematicsYunani| ancientmatematika GreekYunani mathematicskuno]]. [[Euclid]]'s ''[[ Euclid'sElemen ElementsEuklides| ElementsElemen]]'' dari [[Euklides]] ( c. 300 BCSM) proves themembuktikan [[ infinitudebilangan ofprima primestak-hingga]] and thedan [[ fundamentalteorema theoremdasar of arithmeticaritmetika]], anddan showsmenunjukkan howcara to construct amembuat [[ perfectbilangan numbersempurna]] from adari [[ Mersenneprima primeMersenne]].<ref name="stillwell-2010-p40">{{cite book|title=Mathematics and Its History|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|author-link=John Stillwell|edition=3rd|publisher=Springer|year=2010|isbn=978-1-4419-6052-8|page=40|url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40}}</ref> AnotherPenemuan GreekYunani invention,lainnya theyaitu [[ Sieve oftapis Eratosthenes]] , ismasih stilldigunakan useduntuk tomenyusun constructdaftar lists ofbilangan primesprima.<ref name="pomerance-sciam">{{cite journal|title=The Search for Prime Numbers|first=Carl|last=Pomerance|author-link=Carl Pomerance|journal=[[Scientific American]]|volume=247|issue=6|date=December 1982|pages=136–147|jstor=24966751|doi=10.1038/scientificamerican1282-136|bibcode=1982SciAm.247f.136P}}</ref><ref name="mollin">{{cite journal | last = Mollin | first = Richard A. | doi = 10.2307/3219180 | issue = 1 | journal = Mathematics Magazine | mr = 2107288 | pages = 18–29 | title = A brief history of factoring and primality testing B. C. (before computers) | volume = 75 | year = 2002| jstor = 3219180 }}</ref>
Around 1000 AD, the [[Mathematics in medieval Islam|Islamic]] mathematician [[Ibn al-Haytham]] (Alhazen) found [[Wilson's theorem]], characterizing the prime numbers as the numbers <math>n</math> that evenly divide <math>(n-1)!+1</math>. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham|mode=cs1}}</ref> Another Islamic mathematician, [[Ibn al-Banna' al-Marrakushi]], observed that the sieve of Eratosthenes can be sped up by testing only the divisors up to the square root of the largest number to be tested. [[Fibonacci]] brought the innovations from Islamic mathematics back to Europe. His book ''[[Liber Abaci]]'' (1202) was the first to describe [[trial division]] for testing primality, again using divisors only up to the square root.<ref name="mollin"/>
Sekitar 1000 M, matematikawan [[Matematika dalam Islam abad pertengahan|Islam]] [[Ibn al-Haytham]] (Alhazen) menemukan [[teorema Wilson]] dengan mencirikan bilangan prima sebagai bilangan <math>n</math> yang membagi rata <math>(n-1)!+1</math>. Ia juga menduga bahwa semua bilangan sempurna genap berasal dari konstruksi Euklides yang menggunakan bilangan prima Mersenne, tetapi tidak dapat membuktikannya.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham|mode=cs1}}</ref> Matematikawan Islam lainnya, [[Ibn al-Banna' al-Marrakushi]] mengamati bahwa pitas Eratosthenes dapat dipercepat dengan menguji hanya pembagi hingga akar kuadrat dari bilangan terbesar yang akan diuji. [[Fibonacci]] membawa inovasi dari matematika Islam kembali ke Eropa. ''[[Liber Abaci]]'' (1202) dalam bukunya yang pertama mendeskripsikan [[pembagian percobaan]] untuk menguji primalitas, sekali lagi menggunakan pembagi hanya akar kuadrat hingga.<ref name="mollin"/>
In 1640 [[Pierre de Fermat]] stated (without proof) [[Fermat's little theorem]] (later proved by [[Gottfried Wilhelm Leibniz|Leibniz]] and [[Leonhard Euler|Euler]]).<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA45 8. Fermat's Little Theorem (November 2003), p. 45]</ref> Fermat also investigated the primality of the [[Fermat number]]s ▼<math>2^{2^n}+1</math>,<ref>{{cite book|title=How Euler Did Even More|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2014|isbn=978-0-88385-584-3|page=42|url=https://books.google.com/books?id=3c6iBQAAQBAJ&pg=PA42}}</ref> and [[Marin Mersenne]] studied the [[Mersenne prime]]s, prime numbers of the form <math>2^p-1</math> with <math>p</math> itself a prime.<ref>{{cite book|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|publisher=Academic Press|year=2002|isbn=978-0-12-421171-1|page=369|url=https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA369}}</ref> [[Christian Goldbach]] formulated [[Goldbach's conjecture]], that every even number is the sum of two primes, in a 1742 letter to Euler.<ref>{{cite book|title=Goldbach Conjecture|edition=2nd|volume=4|series=Series In Pure Mathematics|first=Wang|last=Yuan|publisher=World Scientific|year=2002|isbn=978-981-4487-52-8|page=21|url=https://books.google.com/books?id=g4jVCgAAQBAJ&pg=PA21}}</ref> Euler proved Alhazen's conjecture (now the [[Euclid–Euler theorem]]) that all even perfect numbers can be constructed from Mersenne primes.<ref name="stillwell-2010-p40"/> He introduced methods from [[mathematical analysis]] to this area in his proofs of the infinitude of the primes and the [[divergence of the sum of the reciprocals of the primes]] <math>\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{5}+\tfrac{1}{7}+\tfrac{1}{11}+\cdots</math>.<ref>{{cite book|title=The Development of Prime Number Theory: From Euclid to Hardy and Littlewood|series=Springer Monographs in Mathematics|first=Wladyslaw|last=Narkiewicz|publisher=Springer|year=2000|isbn=978-3-540-66289-1|page=11|contribution=1.2 Sum of Reciprocals of Primes|contribution-url=https://books.google.com/books?id=VVr3EuiHU0YC&pg=PA11}}</ref> ▼At the start of the 19th century, Legendre and Gauss conjectured that as <math>x</math> tends to infinity, the number of primes up to <math>x</math> is [[Asymptotic analysis|asymptotic]] to <math>x/\log x</math>, where <math>\log x</math> is the [[natural logarithm]] of <math>x</math>. A weaker consequence of this high density of primes was [[Bertrand's postulate]], that for every <math>n > 1</math> there is a prime between <math>n</math> and <math>2n</math>, proved in 1852 by [[Pafnuty Chebyshev]].<ref>{{cite journal|first=P. |last=Tchebychev |author-link=Pafnuty Chebyshev |title=Mémoire sur les nombres premiers. |journal=Journal de mathématiques pures et appliquées |series=Série 1 |year=1852 |pages=366–390 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf |language=fr}}. (Proof of the postulate: 371–382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp. 15–33, 1854</ref> Ideas of [[Bernhard Riemann]] in his [[On the Number of Primes Less Than a Given Magnitude|1859 paper on the zeta-function]] sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related [[Riemann hypothesis]] remains unproven, Riemann's outline was completed in 1896 by [[Jacques Hadamard|Hadamard]] and [[Charles Jean de la Vallée-Poussin|de la Vallée Poussin]], and the result is now known as the [[prime number theorem]].<ref>{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | editor1-last = Bambah | editor1-first = R.P. | editor2-last = Dumir | editor2-first = V.C. | editor3-last = Hans-Gill | editor3-first = R.J. | contribution = A centennial history of the prime number theorem | location = Basel | mr = 1764793 | pages = 1–14 | publisher = Birkhäuser | series = Trends in Mathematics | title = Number Theory | contribution-url = https://books.google.com/books?id=aiDyBwAAQBAJ&pg=PA1 | year = 2000}}</ref> Another important 19th century result was [[Dirichlet's theorem on arithmetic progressions]], that certain [[arithmetic progression]]s contain infinitely many primes.<ref>{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | contribution = 7. Dirichlet's Theorem on Primes in Arithmetical Progressions | contribution-url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA146 | location = New York; Heidelberg | mr = 0434929 | pages = 146–156 | publisher = Springer-Verlag | title = Introduction to Analytic Number Theory | year = 1976 }}</ref> ▼
▲InPada tahun 1640 [[Pierre de Fermat]] statedmenyatakan ( withouttanpa proofpembuktian) [[ Fermat'steorema littlekecil theoremFermat]] , (laterkemudian proveddibuktikan bydengan [[Gottfried Wilhelm Leibniz|Leibniz]] anddan [[Leonhard Euler|Euler]]).<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA45 8. Fermat's Little Theorem (November 2003), phal. 45]</ref> Fermat alsojuga investigatedmenyelidiki the primality of theprimalitas [[ bilangan Fermat number]] s
Many mathematicians have worked on [[primality test]]s for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include [[Pépin's test]] for Fermat numbers (1877),<ref>{{cite book|title=A History of Algorithms: From the Pebble to the Microchip|first=Jean-Luc|last=Chabert|publisher=Springer|year=2012|isbn=978-3-642-18192-4|page=261|url=https://books.google.com/books?id=XcDqCAAAQBAJ&pg=PA261}}</ref> [[Proth's theorem]] (c. 1878),<ref>{{cite book|title=Elementary Number Theory and Its Applications|first=Kenneth H.|last=Rosen|edition=4th|publisher=Addison-Wesley|year=2000|isbn=978-0-201-87073-2|contribution=Theorem 9.20. Proth's Primality Test|page=342}}</ref> the [[Lucas–Lehmer primality test]] (originated 1856), and the generalized [[Lucas primality test]].<ref name="mollin"/> ▼▲<math>2^{2^n}+1</math>,<ref>{{cite book|title=How Euler Did Even More|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2014|isbn=978-0-88385-584-3|page=42|url=https://books.google.com/books?id=3c6iBQAAQBAJ&pg=PA42}}</ref> anddan [[Marin Mersenne]] studied themempelajari [[ prima Mersenne prime]] s, primebilangan numbersprima ofdari the formbentuk <math>2^p-1</math> withdengan <math>p</math> itselfsendiri adalah abilangan primeprima.<ref>{{cite book|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|publisher=Academic Press|year=2002|isbn=978-0-12-421171-1|page=369|url=https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA369}}</ref> [[Christian Goldbach]] formulatedmerumuskan [[ Goldbach'skonjektur conjectureGoldbach]], thatbahwa everysetiap evenbilangan numbergenap isadalah thejumlah sumdari ofdua twobilangan primesprima, indalam asurat 1742tahun letter1742 tountuk Euler.<ref>{{cite book|title=Goldbach Conjecture|edition=2nd|volume=4|series=Series In Pure Mathematics|first=Wang|last=Yuan|publisher=World Scientific|year=2002|isbn=978-981-4487-52-8|page=21|url=https://books.google.com/books?id=g4jVCgAAQBAJ&pg=PA21}}</ref> Euler provedmembuktikan konjektur Alhazen 's conjecture ( nowyang thesaat ini [[ Euclid–Eulerteorema theoremEuklides–Euler]]) thatbahwa semua allbilangan evensempurna perfectgenap numbersdapat candibangun bedari constructedbilangan fromprima Mersenne primes.<ref name="stillwell-2010-p40"/> HeIa introducedmemperkenalkan methodsmetode fromdari [[ mathematicalanalisis analysismatematis]] to thiske area inini hisdalam proofsbuktinya oftentang theketakhinggaan infinitudebilangan ofprima the primes and thedan [[ divergence of the sum ofdivergensi thejumlah reciprocalskebalikan ofdari thebilangan primesprima]] <math>\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{5}+\tfrac{1}{7}+\tfrac{1}{11}+\cdots</math>.<ref>{{cite book|title=The Development of Prime Number Theory: From Euclid to Hardy and Littlewood|series=Springer Monographs in Mathematics|first=Wladyslaw|last=Narkiewicz|publisher=Springer|year=2000|isbn=978-3-540-66289-1|page=11|contribution=1.2 Sum of Reciprocals of Primes|contribution-url=https://books.google.com/books?id=VVr3EuiHU0YC&pg=PA11}}</ref>
▲AtPada theawal startabad of the 19th centuryke-19, Legendre anddan Gauss conjecturedmenduga thatbahwa assebagai <math>x</math> tendscenderung to infinitytak-hingga, the number ofjumlah primesbilangan upprima tohingga <math>x</math> isadalah [[ AsymptoticAnalisis analysisasimtotik| asymptoticasimptotik]] tohingga <math>x/\log x</math>, wheredimana <math>\log x</math> is theadalah [[ naturallogaritma logarithmalami]] ofdari <math>x</math>. AKonsekuensi weakerlemah consequencedari ofkerapatan thisbilangan highprima densitytinggi of primes wasadalah [[ postulat Bertrand 's postulate]], thatbahwa foruntuk everysetiap <math>n > 1</math> thereterdapat isbilangan a primeprima betweendiantara <math>n</math> anddan <math>2n</math> , provedyang indibuktikan pada tahun 1852 byoleh [[Pafnuty Chebyshev]].<ref>{{cite journal|first=P. |last=Tchebychev |author-link=Pafnuty Chebyshev |title=Mémoire sur les nombres premiers. |journal=Journal de mathématiques pures et appliquées |series=Série 1 |year=1852 |pages=366–390 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf |language=fr}}. (Proof of the postulate: 371–382). AlsoLihat seepula Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pphal. 15–33, 1854</ref> Ideas ofGagasan [[Bernhard Riemann]] indalam hiskaryanya [[On the Number of Primes Less Than a Given Magnitude|1859 papermakalah on thetentang fungsi-zeta -function]] sketchedmembuat ansketsa outlinegaris forbesar provinguntuk themembuktikan conjecture ofkonjektur Legendre anddan Gauss. Although the closely relatedMeskipun [[ Riemannhipotesis hypothesisRiemann]] remainsyang terkait erat tetap unproventidak terbukti, garis besar Riemann 's outlinediselesaikan waspada completed intahun 1896 byoleh [[Jacques Hadamard|Hadamard]] anddan [[Charles Jean de la Vallée-Poussin|de la Vallée Poussin]], anddan thehasilnya resultsekarang isdikenal now known as thesebagai [[ primeteorema numberbilangan theoremprima]].<ref>{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | editor1-last = Bambah | editor1-first = R.P. | editor2-last = Dumir | editor2-first = V.C. | editor3-last = Hans-Gill | editor3-first = R.J. | contribution = A centennial history of the prime number theorem | location = Basel | mr = 1764793 | pages = 1–14 | publisher = Birkhäuser | series = Trends in Mathematics | title = Number Theory | contribution-url = https://books.google.com/books?id=aiDyBwAAQBAJ&pg=PA1 | year = 2000}}</ref> AnotherHasil importantterpenting 19thabad centuryke-19 resultlainnya wasadalah [[ Teorema Dirichlet 's theoremtentang onbarisan arithmetic progressionsaritmetika]], that certainbahwa [[ arithmeticbarisan progressionaritmetika]] s containtertentu mengandung banyak infinitelybilangan manyprima primesketakhinggaan.<ref>{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | contribution = 7. Dirichlet's Theorem on Primes in Arithmetical Progressions | contribution-url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA146 | location = New York; Heidelberg | mr = 0434929 | pages = 146–156 | publisher = Springer-Verlag | title = Introduction to Analytic Number Theory | year = 1976 }}</ref>
▲ManyBeberapa mathematiciansmatematikawan havetelah worked onmelakukan [[ primalityuji testprimalitas]] s foruntuk numbersbilangan largerlebih thanbesar thosedari wherebilangan trialpenerapan divisionuji is practicably applicablepembagian. MethodsMetode thatbatasan areuntuk restrictedbentuk tobilangan specifictertentu number forms includetermasuk [[ uji Pépin 's test]] foruntuk bilangan Fermat numbers (1877),<ref>{{cite book|title=A History of Algorithms: From the Pebble to the Microchip|first=Jean-Luc|last=Chabert|publisher=Springer|year=2012|isbn=978-3-642-18192-4|page=261|url=https://books.google.com/books?id=XcDqCAAAQBAJ&pg=PA261}}</ref> [[Proth's theorem]] (c. 1878),<ref>{{cite book|title=Elementary Number Theory and Its Applications|first=Kenneth H.|last=Rosen|edition=4th|publisher=Addison-Wesley|year=2000|isbn=978-0-201-87073-2|contribution=Theorem 9.20. Proth's Primality Test|page=342}}</ref> the [[ Lucas–Lehmeruji primalityprimalitas testLucas–Lehmer]] ( originateddari tahun 1856), anddan the generalizedgeneralisasi [[ Lucasuji primalityprimalitas testLucas]].<ref name="mollin"/>
Since 1951 all the [[largest known prime]]s have been found using these tests on [[computer]]s.{{efn|A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.<ref>{{cite book|title=The Once and Future Turing|first1=S. Barry|last1=Cooper|first2=Andrew|last2=Hodges|publisher=Cambridge University Press|year=2016|isbn=978-1-107-01083-3|pages=37–38|url=https://books.google.com/books?id=h12cCwAAQBAJ&pg=PA37}}</ref>}} The search for ever larger primes has generated interest outside mathematical circles, through the [[Great Internet Mersenne Prime Search]] and other [[distributed computing]] projects.<ref name=ziegler/><ref>{{harvnb|Rosen|2000}}, p. 245.</ref> The idea that prime numbers had few applications outside of [[pure mathematics]]{{efn|name="pure"|For instance, Beiler writes that number theorist [[Ernst Kummer]] loved his [[ideal number]]s, closely related to the primes, "because they had not soiled themselves with any practical applications",<ref>{{cite book|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|first=Albert H.|last=Beiler|year=1999|publisher=Dover|orig-year=1966|isbn=978-0-486-21096-4|page=2|url=https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA2|oclc=444171535}}</ref> and Katz writes that [[Edmund Landau]], known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as [[geometry]] that had already shown themselves to be useful.<ref>{{cite journal | last = Katz | first = Shaul | doi = 10.1017/S0269889704000092 | issue = 1–2 | journal = Science in Context | mr = 2089305 | pages = 199–234 | title = Berlin roots – Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem | volume = 17 | year = 2004| s2cid = 145575536 }}</ref>}} was shattered in the 1970s when [[public-key cryptography]] and the [[RSA (cryptosystem)|RSA]] cryptosystem were invented, using prime numbers as their basis.<ref name="ent-7">{{cite book|title=Elementary Number Theory|series=Textbooks in mathematics|first1=James S.|last1=Kraft|first2=Lawrence C.|last2=Washington|publisher=CRC Press|year=2014|isbn=978-1-4987-0269-0|page=7|url=https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA7}}</ref> ▼
▲SinceSejak tahun 1951 , all thesemua [[ largestbilangan knownprima primeterbesar yang diketahui]] s havetelah beenditemukan foundmenggunakan usingtes theseini tests onpada [[ computerkomputer]] s.{{efn| ASebuah bilangan prima 44-digit primeyang numberditemukan foundpada intahun 1951 byoleh Aimé Ferrier withdengan akalkulator mechanicalmekanik calculatortetap remainsmerupakan thebilangan largestprima primeterbesar notyang totidak haveditemukan beendengan found with the aid ofbantuan electronickomputer computerselektronik.<ref>{{cite book|title=The Once and Future Turing|first1=S. Barry|last1=Cooper|first2=Andrew|last2=Hodges|publisher=Cambridge University Press|year=2016|isbn=978-1-107-01083-3|pages=37–38|url=https://books.google.com/books?id=h12cCwAAQBAJ&pg=PA37}}</ref>}} ThePencarian searchbilangan forprima everbesar largertelah primesmembangkitkan hasminat generatedpada interestluar outsidelingkaran mathematical circlesmatematika, through themelalui [[ GreatPencarian InternetUtama Mersenne PrimeInternet SearchHebat]] anddan otherproyek [[ distributedkomputasi computingdistribusi]] projectslainnya.<ref name=ziegler/><ref>{{harvnb|Rosen|2000}}, phal. 245.</ref> TheGagasan ideabahwa thatbilangan primeprima numbersmemiliki hadbeberapa fewaplikasi applications outside ofdiluar [[ purematematika mathematicsmurni]] ,{{efn|name="pure"| For instanceMisalnya, Beiler writesmenulis thatbahwa numberahli theoristteori bilangan [[Ernst Kummer]] loved hismenyukai [[ bilangan ideal number]] s miliknya, closelyyang relatedterkait toerat thedengan bilangan primesprima, " becausekarena theymereka hadtidak notmengotori soileddiri themselvesmereka withdengan anyaplikasi practicalpraktis applicationsapa pun",<ref>{{cite book|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|first=Albert H.|last=Beiler|year=1999|publisher=Dover|orig-year=1966|isbn=978-0-486-21096-4|page=2|url=https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA2|oclc=444171535}}</ref> andbahkan Katz writesmenulis thatbahwa [[Edmund Landau]] , knownyang fordikenal hiskarena workkaryanya ontentang thedistribusi distributionbilangan ofprima primes,yaitu " ''loathed practical applications of mathematics ''" , anddan foruntuk thisalasan reasontersebut avoideduntuk subjectsmenghindari suchsubjek asseperti [[ geometrygeometri]] that had already shown themselvesyang totelah beterbukti usefulberguna.<ref>{{cite journal | last = Katz | first = Shaul | doi = 10.1017/S0269889704000092 | issue = 1–2 | journal = Science in Context | mr = 2089305 | pages = 199–234 | title = Berlin roots – Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem | volume = 17 | year = 2004| s2cid = 145575536 }}</ref>}} wassekitar shatteredtahun in the 1970s1970-an whenketika [[ public-keykriptografi kunci cryptographypublik]] and thedan [[RSA ( cryptosystemsistem kripto)|RSA]] cryptosystemsistem werekripto invented,ditemukan usingdengan primemenggunakan numbersbilangan asprima theirsebagai basisbasisnya.<ref name="ent-7">{{cite book|title=Elementary Number Theory|series=Textbooks in mathematics|first1=James S.|last1=Kraft|first2=Lawrence C.|last2=Washington|publisher=CRC Press|year=2014|isbn=978-1-4987-0269-0|page=7|url=https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA7}}</ref>
The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form.<ref name="pomerance-sciam"/><ref>{{cite book|title=Secret History: The Story of Cryptology|series=Discrete Mathematics and Its Applications|first=Craig P.|last=Bauer|publisher=CRC Press|year=2013|isbn=978-1-4665-6186-1|page=468|url=https://books.google.com/books?id=EBkEGAOlCDsC&pg=PA468}}</ref><ref>{{cite book|title=Old and New Unsolved Problems in Plane Geometry and Number Theory|volume=11|series=Dolciani mathematical expositions|first1=Victor|last1=Klee|author1-link=Victor Klee|first2=Stan|last2=Wagon|author2-link=Stan Wagon|publisher=Cambridge University Press|year=1991|isbn=978-0-88385-315-3|page=224|url=https://books.google.com/books?id=tRdoIhHh3moC&pg=PA224}}</ref> The mathematical theory of prime numbers also moved forward with the [[Green–Tao theorem]] (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and [[Yitang Zhang]]'s 2013 proof that there exist infinitely many [[prime gap]]s of bounded size.<ref name="neale-18-47">{{harvnb|Neale|2017}}, pp. 18, 47.</ref> ▼
▲TheMeningkatnya increasedkepentingan practicalpraktis importancedari ofpengujian computerizeddan primalityfaktorisasi testingprimalitas andterkomputerisasi factorizationmenyebabkan ledpengembangan tometode themenjadi developmentlebih ofbaik improvedyang methodsmampu capablemenangani ofsejumlah handling large numbers ofbesar unrestrictedbentuk formketakhinggaan.<ref name="pomerance-sciam"/><ref>{{cite book|title=Secret History: The Story of Cryptology|series=Discrete Mathematics and Its Applications|first=Craig P.|last=Bauer|publisher=CRC Press|year=2013|isbn=978-1-4665-6186-1|page=468|url=https://books.google.com/books?id=EBkEGAOlCDsC&pg=PA468}}</ref><ref>{{cite book|title=Old and New Unsolved Problems in Plane Geometry and Number Theory|volume=11|series=Dolciani mathematical expositions|first1=Victor|last1=Klee|author1-link=Victor Klee|first2=Stan|last2=Wagon|author2-link=Stan Wagon|publisher=Cambridge University Press|year=1991|isbn=978-0-88385-315-3|page=224|url=https://books.google.com/books?id=tRdoIhHh3moC&pg=PA224}}</ref> TheTeori mathematicalmatematika theorybilangan ofprima primejuga numbersterus alsoberkembang moved forward with thedengan [[ Green–Taoteorema theoremGreen-Tao]] (2004) thatbahwa therebarisan arearitmetika arbitrarilypanjang longyang arithmeticcenderung progressionsdari ofbilangan prime numbersprima, anddan pembuktian pada tahun 2013 [[Yitang Zhang]] 's 2013bahwa proofmemiliki that there exist infinitely manybanyak [[ primeuji gapcelah prima]] s of bounded sizeketakhinggaan.<ref name="neale-18-47">{{harvnb|Neale|2017}}, pp. 18, 47.</ref>
<!--
===Primality of one===
MostKebanyakan earlyorang GreeksYunani didawal notbahkan eventidak considermenganggap 1 tosebagai be a numberangka,<ref name="crxk-34">{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Reddick | first2 = Angela | last3 = Xiong | first3 = Yeng | last4 = Keller | first4 = Wilfrid | issue = 9 | journal = [[Journal of Integer Sequences]] | mr = 3005523 | page = Article 12.9.8 | title = The history of the primality of one: a selection of sources | url = https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html | volume = 15 | year = 2012 }} For a selection of quotes from and about the ancient Greek positions on this issue, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6.</ref><ref>{{cite book|title=Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary|volume=39|series=Philosophia Antiqua : A Series of Monographs on Ancient Philosophy|first=Leonardo|last=Tarán|publisher=Brill|year=1981|isbn=978-90-04-06505-5|pages=35–38|url=https://books.google.com/books?id=cUPXqSb7V1wC&pg=PA35}}</ref> so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The [[Mathematics in medieval Islam|medieval Islamic mathematicians]] largely followed the Greeks in viewing 1 as not being a number.<ref name="crxk-34"/>
By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number.<ref>{{harvnb|Caldwell|Reddick|Xiong|Keller|2012}}, pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet.</ref> In the mid-18th century [[Christian Goldbach]] listed 1 as prime in his correspondence with [[Leonhard Euler]]; however, Euler himself did not consider 1 to be prime.<ref>{{harvnb|Caldwell|Reddick|Xiong|Keller|2012}}, p. 15.</ref> In the 19th century many mathematicians still considered 1 to be prime,<ref name="cx"/> and lists of primes that included 1 continued to be published as recently as 1956.<ref>{{cite book | last=Riesel | first=Hans | author-link= Hans Riesel | title=Prime Numbers and Computer Methods for Factorization | publisher=Birkhäuser | location=Basel, Switzerland | isbn=978-0-8176-3743-9 | year=1994|page=36|edition=2nd|mr=1292250|url=https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA36 | doi=10.1007/978-1-4612-0251-6 }}</ref><ref name="cg-bon-129-130">{{cite book | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Guy | first2=Richard K. | author2-link=Richard K. Guy | title=The Book of Numbers | url=https://archive.org/details/bookofnumbers0000conw | url-access=registration | publisher=Copernicus | location=New York | isbn=978-0-387-97993-9 | year=1996 | pages = [https://archive.org/details/bookofnumbers0000conw/page/129 129–130] | mr=1411676 | doi=10.1007/978-1-4612-4072-3 }}</ref>
If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1.<ref name="cx">{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Xiong | first2 = Yeng | issue = 9 | journal = [[Journal of Integer Sequences]] | mr = 3005530 | page = Article 12.9.7 | title = What is the smallest prime? | url = https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf | volume = 15 | year = 2012}}</ref> Similarly, the [[sieve of Eratosthenes]] would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1.<ref name="cg-bon-129-130"/> Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for [[Euler's totient function]] or for the [[Sum-of-divisors function|sum of divisors function]] are different for prime numbers than they are for 1.<ref>For the totient, see {{harvnb|Sierpiński|1988}}, [https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA245 p. 245]. For the sum of divisors, see {{cite book|title=How Euler Did It|series=MAA Spectrum|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2007|isbn=978-0-88385-563-8|page=59|url=https://books.google.com/books?id=sohHs7ExOsYC&pg=PA59}}</ref> By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "[[Unit (ring theory)|unit]]".<ref name="cx"/>-->
== Sifat dasar ==
Meskipun konjektur telah dirumuskan tentang proporsi bilangan prima dalam polinomial tingkat tinggi mereka tetap tidak terbukti, dan tidak diketahui apakah polinomial kuadrat (untuk argumen bilangan bulat) adalah bilangan prima tak-hingga.
===Bukti analitik teorema Euklides===
===Analytical proof of Euclid's theorem===
[[Bukti bahwa jumlah kebalikan dari bilangan prima divergen|Bukti Euler bahwa ada banyak bilangan prima]] yang mempertimbangkan jumlah [[perkalian invers|kebalikan]] dari bilangan prima
[[Proof that the sum of the reciprocals of the primes diverges|Euler's proof that there are infinitely many primes]] considers the sums of [[Multiplicative inverse|reciprocals]] of primes,
:<math>\frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p.</math>
Euler showedmenunjukkan that,bahwa foruntuk anynilai arbitrarysembarang [[bilangan real number]] <math>x</math>, theremenunjukkan existssebuah abilangan primeprima <math>p</math> foryang whichjumlahnya thislebih sumbesar is bigger thandari <math>x</math>. <ref>{{harvnb|Apostol|1976}}, SectionBagian 1.6, TheoremTeorema 1.13</ref> ThisHal showstersebut thatmenunjukkan therebahwa areada infinitelybanyak manybilangan primesprima karena ada banyak bilangan prima, becausejumlah iftersebut thereakan weremencapai finitelynilai manymaksimumnya primespada thebilangan sumprima wouldterbesar reachdari itssetiap maximummelewati value<math>x</math>. atTingkat thepertumbuhan biggestjumlah primedapat ratherdijelaskan thanlebih growingtepat pastoleh every[[Teorema Mertens|teorema kedua Mertens]].<mathref>x{{harvnb|Apostol|1976}}, Bagian 4.8, Teorema 4.12</mathref>. Sebagai perbandingan, jumlah
The growth rate of this sum is described more precisely by [[Mertens' theorems|Mertens' second theorem]].<ref>{{harvnb|Apostol|1976}}, Section 4.8, Theorem 4.12</ref> For comparison, the sum
:<math>\frac 1 {1^2} + \frac 1 {2^2} + \frac 1 {3^2} + \cdots + \frac 1 {n^2}</math>
althoughtidak bothtumbuh setssecara areketakhinggaan infiniteapabila <math>n</math> menjadi sebagai ketakhinggaan (lihat [[masalah Basel]]). Dalam pengertian ini, bilangan prima sering muncul dibandingkan bilangan kuadrat asli, meskipun kedua himpunan tersebut adalah ketakhinggaan.<ref name="mtb-invitation">{{cite book|title=An Invitation to Modern Number Theory|first1=Steven J.|last1=Miller|first2=Ramin|last2=Takloo-Bighash|publisher=Princeton University Press|year=2006|isbn=978-0-691-12060-7|pages=43–44|url=https://books.google.com/books?id=kLz4z8iwKiwC&pg=PA43}}</ref> [[ Brun'sTeorema theoremBrun]] statesmenyatakan thatbahwa thejumlah sumkebalikan of the reciprocals ofdari [[ twinprima primekembar]] s,▼does not grow to infinity as <math>n</math> goes to infinity (see the [[Basel problem]]). In this sense, prime numbers occur more often than squares of natural numbers,
▲although both sets are infinite.<ref name="mtb-invitation">{{cite book|title=An Invitation to Modern Number Theory|first1=Steven J.|last1=Miller|first2=Ramin|last2=Takloo-Bighash|publisher=Princeton University Press|year=2006|isbn=978-0-691-12060-7|pages=43–44|url=https://books.google.com/books?id=kLz4z8iwKiwC&pg=PA43}}</ref> [[Brun's theorem]] states that the sum of the reciprocals of [[twin prime]]s,
:<math> \left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1}{{11}} + \frac{1}{{13}}} \right) + \cdots, </math>
ismerupakan finite.berhingga, Becausekarena ofteorema Brun's theorem,tidak itmungkin ismenggunakan not possible to usemetode Euler's methoduntuk to solve themenyelesaikan [[twinkonjektur primeprima conjecturekembar]], thatbahwa masih thereada existbanyak infinitelybilangan manyprima twinkembar primesketakhinggaan.<ref name="mtb-invitation"/>
===Jumlah bilangan prima bawah batas yang diberikan ===
===Number of primes below a given bound===
{{Main|Teorema bilangan prima|Fungsi pencacahan prima}}
{{Main|Prime number theorem|Prime-counting function}}
[[FileBerkas:Prime-counting relative error.svg|thumb|upright=1.6|The [[ApproximationKesalahan errorperkiraan|relativeKesalahan errorrelatif]] ofdari <math>\tfrac{n}{\log n}</math> anddan theintegral logarithmic integrallogaritmik <math>\operatorname{Li}(n)</math> assebagai approximationsaproksimasi to theuntuk [[prime-countingfungsi pencacahan functionprima]]. BothKedua relativekesalahan errorsrelatif decreaseberkurang tomenjadi zeronol asketika <math>n</math> growstumbuh, buttetapi thekonvergensi convergenceke tonol zerojauh islebih muchcepat moreuntuk rapidintegral for the logarithmic integrallogaritmik.]]
The [[prime-countingFungsi functionpencacahan prima]] <math>\pi(n)</math> is defineddidefinisikan assebagai thejumlah numberbilangan ofprima primestidak notlebih greaterbesar thandari <math>n</math>.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA6 p. 6].</ref> For example,Misalnya <math>\pi(11)=5</math>, sincekarena thereada arelima fivebilangan primesprima lessyang thankurang ordari equalatau tosama dengan 11. MethodsMetode suchyang as theseperti [[Meissel–Lehmeralgoritma algorithmMeissel–Lehmer]] canmenghitung computenilai exact values ofeksak <math>\pi(n)</math> faster than itlebih wouldcepat bedibandingkan possibleuntuk tomembuat listdaftar eachsetiap primebilangan upprima tohingga <math>n</math>.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA152 Section 3.7, CountingPencacahan primesprima, ppham. 152–162152-162].</ref> The [[primeTeorema numberbilangan theoremprima]] statesmenyatakan thatbahwa <math>\pi(n)</math> isadalah asymptoticasimtotik topada <math>n/\log n</math>, which isyang denoteddinotasikan assebagai
: <math>\pi(n) \sim \frac{n}{\log n},</math>
anddan meansberarti that the ratio ofrasio <math>\pi(n)</math> toterhadap thepecahan right-hand fractionkanan [[convergentbarisan sequencekonvergen|approachespendekatan]] 1 asketika <math>n</math> growsbertambah tosecara infinityketakhinggaan.<ref name="cranpom10">{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA10 phal. 10].</ref> ThisHal impliesini thatdisebutkan thebahwa likelihoodkemungkinan thatbahwa abilangan randomlyyang chosendipilih numbersecara lessacak thankurang dari <math>n</math> isadalah primebilangan isprima (approximatelykurang-lebih) inversely proportional toberbanding thekebalikan numberdengan ofjumlah digitsdigit indalam <math>n</math>.<ref>{{cite book|title=The Number Mysteries: A Mathematical Odyssey through Everyday Life|first=Marcus|last=du Sautoy|author-link=Marcus du Sautoy|publisher=St. Martin's Press|year=2011|isbn=978-0-230-12028-0|pages=50–52|contribution=What are the odds that your telephone number is prime?|contribution-url=https://books.google.com/books?id=snaUbkIb8SEC&pg=PA50}}</ref>
ItHal alsotersebut impliesjuga thatmenyebutkan thebahwa bilangan prima ke-<math>n</math>th prime number is proportionalberbanding todengan <math>n\log n</math><ref>{{harvnb|Apostol|1976}}, SectionBagian 4.6, TheoremTeorema 4.7</ref>
anddan thereforeoleh thatkarena theitu averageukuran size of a primerata-rata gapcelah isprima proportionalberbanding todengan <math>\log n</math>.<ref name="riesel-gaps">{{harvnb|Riesel|1994}}, "[https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA78 Large gaps between consecutive primes]", pphal. 78–79.</ref>
AEstimasi moreyang accurateakurat estimate foruntuk <math>\pi(n)</math> is given bydiberikan theoleh [[offsetofset logarithmiclogaritmik integral]]<ref name="cranpom10"/>
:<math>\pi(n)\sim \operatorname{Li}(n) = \int_2^n \frac{dt}{\log t}.</math>
===ArithmeticBarisan progressionsaritmetika===
{{main|Teorema Dirichlet's theoremtentang onbarisan arithmetic progressionsaritmatika|Green–TaoTeorema theoremGreen–Tao}}
AnSebuah [[arithmeticbarisan progressionaritmetika]] isadalah abarisan finitebilangan orberhingga infiniteatau sequenceketakhinggaan ofsehingga numbers such that consecutivebilangan-bilangan numbersberurutan indalam thebarisan sequencetersebut allsemuanya havememiliki theselisih sameyang differencesama.<ref>{{cite book|title=Algebra|first1=I.M.|last1=Gelfand|author1-link=Israel Gelfand|first2=Alexander|last2=Shen|publisher=Springer|year=2003|isbn=978-0-8176-3677-7|page=37|url=https://books.google.com/books?id=Z9z7iliyFD0C&pg=PA37}}</ref> ThisPerbedaan differenceini is called thedisebut [[ModularAritmetika arithmeticmodular|modulus]] ofdari the progressionbarisan.<ref>{{cite book|title=Fundamental Number Theory with Applications|series=Discrete Mathematics and Its Applications|first=Richard A.|last=Mollin|publisher=CRC Press|year=1997|isbn=978-0-8493-3987-5|page=76|url=https://books.google.com/books?id=Fsaa3MUUQYkC&pg=PA76}}</ref> For exampleMisalnya,
:3, 12, 21, 30, 39, ...,
yang adalah barisan aritmetika ketakhinggaan dengan modulus 9. Dalam barisan aritmetika, semua bilangan memiliki sisa yang sama jika dibagi dengan modulus; contohnya, sisanya adalah 3. Karena modulus 9 dan sisanya 3 adalah kelipatan 3, demikian pula setiap elemen dalam barisan. Oleh karena itu, barisan ini hanya berisi satu bilangan prima, 3 itu sendiri. Secara umum, barisan ketakhinggaan-nya
is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression
:<math>a, a+q, a+2q, a+3q, \dots</math>
canmemiliki havelebih moredari thansatu onebilangan primeprima onlyhanya whenbila its remaindersisa <math>a</math> anddan modulus <math>q</math> arerelatif relatively primeprima. IfApabila theymereka arerelatif relatively primeprima, [[Dirichlet'steorema theoremDirichlet ontentang arithmeticbarisan progressionsaritmatika]] assertsmenyatakan thatbahwa thebarisan progressiontersebut containsmemiliki infinitelybanyak manybilangan primesprima.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA Theorem 1.1.5, p. 12].</ref>
{{Wide image|Prime numbers in arithmetic progression mod 9 zoom in.png|815px|PrimesBilangan inprima thedalam arithmeticbarisan progressionsaritmetika modulo 9. EachSetiap rowbaris of the thinpita horizontal bandtipis showsmenunjukkan onesalah ofsatu thedari ninesembilan possiblekemungkinan progressionsbarisan mod 9, withdengan primebilangan numbersprima markedditandai indengan redwarna merah. The progressions of numbers thatBarisan arebilangan 0, 3, oratau 6 mod 9 containmengandung atpaling mostbanyak onesatu primebilangan numberprima (the numberbilangan 3); the remaining progressions of numberssisa thatbarisan arebilangan 2, 4, 5, 7, anddan 8 mod 9 have infinitelymemiliki manybanyak primebilangan numbersprima, withdengan similarbilangan numbersprima ofyang primessama inpada eachsetiap progressionbarisannya|alt=PrimeBilangan numbersprima indalam arithmeticbarisan progressionaritmetika mod 9.}}
The [[Teorema Green–Tao theorem]] shows thatmenunjukkan therebahwa areada arbitrarilybarisan longaritmetika finiteberhingga arithmeticyang progressionspanjangnya consistingterdiri onlydari ofbilangan primesprima.<ref name="neale-18-47"/><ref>{{cite journal|first1=Ben|last1=Green|author1-link=Ben J. Green|first2=Terence|last2=Tao|author2-link=Terence Tao|title=The primes contain arbitrarily long arithmetic progressions|journal=[[Annals of Mathematics]]|volume=167|issue=2|year=2008|pages=481–547|doi=10.4007/annals.2008.167.481|arxiv=math.NT/0404188|s2cid=1883951}}</ref>
===Nilai bilangan prima polinomial pada kuadrat===
===Prime values of quadratic polynomials===
[[FileBerkas:Ulam 2.png|thumb|upright=1.1|The [[Spiral Ulam spiral]]. PrimeBilangan numbersprima (redmerah) clustermengelompokkan onpada somebeberapa diagonalsdiagonal anddan nottidak otherspada yang lain. PrimeNilai valuesbilangan ofprima dari <math>4n^2 - 2n + 41</math> areditampilkan showndengan inwarna bluebiru.|alt=TheSpiral Ulam spiral]]
Euler notedmencatat thatbahwa the functionfungsi
:<math>n^2 - n + 41</math>
yieldsmenghasilkan primebilangan numbersprima foruntuk <math>1\le n\le 40</math>, althoughmeskipun compositebilangan numberskomposit appearmuncul amongantara itsnilai-nilai later valuesselanjutnya.<ref>{{cite book|title=Additive Theory of Prime Numbers|last1=Hua|first1=L.K.|publisher=American Mathematical Society|year=2009|isbn=978-0-8218-4942-2|series=Translations of Mathematical Monographs|volume=13|location=Providence, RI|pages=176–177|mr=0194404|oclc=824812353|orig-year=1965}}</ref><ref>TheUrutan sequence ofbilangan theseprima primesini, startingdimulai atdari <math>n=1</math> ratherdan thanbukan <math>n=0</math>, is listedditulis byoleh {{cite book|title=103 curiosità matematiche: Teoria dei numeri, delle cifre e delle relazioni nella matematica contemporanea|language=it|first1=Paolo Pietro|last1=Lava|first2=Giorgio|last2=Balzarotti|publisher=Ulrico Hoepli Editore S.p.A.|year=2010|isbn=978-88-203-5804-4|page=133|contribution-url=https://books.google.com/books?id=YfsSAAAAQBAJ&pg=PA133|contribution=Chapter 33. Formule fortunate}}</ref> ThePencarian searchpenjelasan foruntuk anfenomena explanationini formengarah this phenomenon led to the deeppada [[algebraicteori numberbilangan theoryaljabar]] ofyang mendalam dari [[bilangan Heegner number]]s and thedan [[classmasalah numberbilangan problemkelas]].<ref>{{cite book|title=Single Digits: In Praise of Small Numbers|first=Marc|last=Chamberland|publisher=Princeton University Press|year=2015|isbn=978-1-4008-6569-7|contribution=The Heegner numbers|pages=213–215|contribution-url=https://books.google.com/books?id=n9iqBwAAQBAJ&pg=PA213}}</ref> The [[Konjektur F Hardy-Littlewood conjecture F]] predictsmemprediksi thekerapatan densitybilangan ofprima primesantara among the values ofnilai-nilai [[quadraticpolinomial polynomialkuadrat]]s withdengan integerbilangan bulat [[coefficientkoefisien]]s dalam hal integral logaritmik dan polinomial. Tidak ada polinomial kuadrat yang terbukti memiliki banyak nilai prima secara ketakhinggaan.<ref name="guy-a1">{{cite book|title=Unsolved Problems in Number Theory|series=Problem Books in Mathematics|edition=3rd|first=Richard|last=Guy|author-link=Richard K. Guy|publisher=Springer|year=2013|isbn=978-0-387-26677-0|pages=7–10|contribution-url=https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA7|contribution=A1 Prime values of quadratic functions}}</ref>
in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.<ref name="guy-a1">{{cite book|title=Unsolved Problems in Number Theory|series=Problem Books in Mathematics|edition=3rd|first=Richard|last=Guy|author-link=Richard K. Guy|publisher=Springer|year=2013|isbn=978-0-387-26677-0|pages=7–10|contribution-url=https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA7|contribution=A1 Prime values of quadratic functions}}</ref>
[[Spiral Ulam]] mengatur bilangan asli dalam kisi dua dimensi, berputar dalam kotak konsentris biasanya mengelilingi asal dengan bilangan prima yang disorot. Secara visual, bilangan prima tampak mengelompokkan pada diagonal tertentu dan bukan pada diagonal lainnya, hal itu menunjukkan bahwa beberapa polinomial kuadrat mengambil nilai prima lebih sering dibanding yang lainnya.<ref name="guy-a1"/>
The [[Ulam spiral]] arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.<ref name="guy-a1"/>
===Fungsi Zeta functiondan and thehipotesis Riemann hypothesis===
{{Main|RiemannHipotesis hypothesisRiemann}}
[[FileBerkas:Riemann zeta function absolute value.png|thumb|upright=1.5|Plot ofnilai theabsolut absolute values ofdari thefungsi zeta function, showingmenunjukkan somebeberapa of its featuresfiturnya|alt=Plot ofnilai theabsolut absolutedari values of thefungsi zeta function]]
OneSalah ofsatu thepertanyaan mosttak famousterpecahkan unsolvedpaling questionsterkenal indalam mathematics,matematika berasal datingdari fromtahun 1859, anddan onesalah ofsatunya thedari [[MillenniumMasalah PrizeHadiah ProblemsMilenium]], is theadalah [[Hipotesis Riemann hypothesis]], which asks whereyang thedimana [[zeronol of a functionfungsi|zerosnol]] of thedari [[fungsi Riemann zeta function]] <math>\zeta(s)</math> are locatedberada.
ThisFungsi functionini is anadalah [[analyticfungsi functionanalitik]] on thepada [[complexbilangan numberkompleks]]s. ForUntuk complexbilangan numberskompleks <math>s</math> withdengan bagian real partlebih greaterbesar thandari onesatu itsama equals both andengan [[seriesderet (mathematicsmatematika)|infinitejumlah sumketakhinggaan]] overpada allsemua integers,bilangan andbulat andan [[infinitedarab producttak hingga|darab ketakhinggaan]] over theatas primebilangan numbers,prima
:<math>\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ primeprima}} \frac 1 {1-p^{-s}}.</math>
ThisKesetaraan equalityantara betweenjumlah adan sumdarab andditemukan a product, discovered byoleh Euler, isyang calleddisebut anjuga [[darab Euler product]].<ref>{{cite book | last = Patterson | first = S.J. | doi = 10.1017/CBO9780511623707 | isbn = 978-0-521-33535-5 | mr = 933558 | page = 1 | publisher = Cambridge University Press, Cambridge | series = Cambridge Studies in Advanced Mathematics | title = An introduction to the theory of the Riemann zeta-function | url = https://books.google.com/books?id=IdHLCgAAQBAJ&pg=PA1 | volume = 14 | year = 1988 }}</ref> TheDarab Euler productdapat canditentukan bedari derivedteorema fromdasar thearitmetika fundamentaldan theoremmenunjukkan ofrelasi arithmetic,erat andantara shows the close connection between thefungsi zeta function and thedan primebilangan numbersprima.<ref>{{cite book | last1 = Borwein | first1 = Peter | author1-link = Peter Borwein | last2 = Choi | first2 = Stephen | last3 = Rooney | first3 = Brendan | last4 = Weirathmueller | first4 = Andrea | doi = 10.1007/978-0-387-72126-2 | isbn = 978-0-387-72125-5 | location = New York | mr = 2463715 | pages = 10–11 | publisher = Springer | series = CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC | title = The Riemann hypothesis: A resource for the afficionado and virtuoso alike | url = https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA10 | year = 2008 }}</ref>
Hal ini pula mengarah pada bukti lain bahwa banyak bilangan prima berhingga: jikalau memiliki banyak berhingga, maka persamaan jumlah-darb juga akan berlaku pada <math>s=1</math>, tetapi jumlahnya akan berbeda ([[Deret harmonik (matematika)|deret harmonik]] <math>1+\tfrac{1}{2}+\tfrac{1}{3}+\dots</math>) ketika darab berhingga kontradiksi.<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA191 hal. 191–193].</ref>
It leads to another proof that there are infinitely many primes: if there were only finitely many,
then the sum-product equality would also be valid at <math>s=1</math>, but the sum would diverge (it is the [[Harmonic series (mathematics)|harmonic series]] <math>1+\tfrac{1}{2}+\tfrac{1}{3}+\dots</math>) while the product would be finite, a contradiction.<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA191 pp. 191–193].</ref>
TheHipotesis Riemann hypothesismenyebutkan statesbahwa that thefungsi-zeta [[zeronol of a functionfungsi|zerosnol]] ofdari thesemua zeta-functionadalah arebilangan allgenap eithernegatif negativeatau evenbilangan numbers, or complex numberskompleks withdengan [[realbagian partreal]] equalsama todengan 1/2.<ref>{{harvnb|Borwein|Choi|Rooney|Weirathmueller|2008}}, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA15 ConjectureKonjektur 2.7 (thehipotesis Riemann hypothesis), phal. 15].</ref> Bukti Thesebenarnya original proof of thedari [[primeteorema numberbilangan theoremprima]] waspada baseddasarnya onadalah abentuk weaklemah formdari of this hypothesishipotesis, thatnamun theretidak areada nonol zerosdengan withbagian real part equalsama todengan 1,<ref>{{harvnb|Patterson|1988}}, p. 7.</ref><ref name="bcrw18">{{harvnb|Borwein|Choi|Rooney|Weirathmueller|2008}}, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA18 p. 18.]</ref> althoughmeskipun otherbukti morelain elementaryyang proofslebih havemendasar beentelah foundditemukan.<ref>{{harvnb|Nathanson|2000}}, [https://books.google.com/books?id=sE7lBwAAQBAJ&pg=PA289 ChapterBab 9, The primeTeorema numberbilangan theoremprima, pphal. 289–324].</ref>
TheFungsi prime-countinghitung functionbilangan canprima bedisebutkan expressed bydengan [[Riemann'srumus expliciteksplisit formulaRiemann]] assebagai ajumlah sumyang indimana whichsetiap eachsuku termberasal comesdari fromsalah onesatu ofnol thedari zeros of thefungsi zeta, function;suku theutama maindari termjumlah ofini this sum is the logarithmicadalah integral, andlogaritma thedan remaining terms causesuku-suku thetersebut sumlainnya tomenghasilkan fluctuatejumlah aboveberfluktuasi andatas belowdan thebawah mainsuku termutama.<ref>{{cite journal | last = Zagier | first = Don | author-link = Don Zagier | doi = 10.1007/bf03351556 | issue = S2 | journal = [[The Mathematical Intelligencer]] | pages = 7–19 | title = The first 50 million prime numbers | volume = 1 | year = 1977| s2cid = 37866599 }} SeeLihat especiallykhususnya pphal. 14–16.</ref>
Dalam pengertian ini, nol mengontrol beberapa bilangan prima reguler distribusi. Apabila hipotesis Riemann tersebut benar, maka fluktuasi akan kecil dan [[distribusi asimtotik]] bilangan prima yang diberikan oleh teorema bilangan prima juga bertahan pada interval yang jauh lebih pendek (panjangnya sekitar akar kuadrat dari <math>x</math> untuk interval yang dekat dengan bilangan <math>x</math>).<ref name="bcrw18"/>
In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the
[[asymptotic distribution]] of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of <math>x</math> for intervals near a number <math>x</math>).<ref name="bcrw18"/>
==Abstract algebra==
===ModularAritmetika arithmeticmodular anddan finiteMedan fieldsberhingga===
{{Main|ModularAritmetika arithmeticmodular}}
ModularAritmetika arithmeticmodular modifiesmemodifikasi usualaritmetika arithmeticbiasa, byhanya onlysaja usingdengan themenggunakan numbersbilangan <math>\{0,1,2,\dots,n-1\}</math>, for auntuk naturalbilangan numberasli <math>n</math> calledyang thedisebut modulus.
AnyBilangan otherasli naturallainnya numberdapat candipetakan beke mappeddalam intosistem thisini systemdengan bymenggantinya replacingdengan itsisa bysetelah itspembagian remainder after division bydengan <math>n</math>.<ref>{{harvtxt|Kraft|Washington|2014}}, [https://books.google.com/books?id=VG9YBQAAQBAJ&pg=PA96 PropositionProposisi 5.3], phal. 96.</ref>
onJumlah, thepembeda, resultdan ofdarab themodular usualdihitung sumdengan melakukan penggantian yang sama dengan sisa hasil penjumlahan, differenceselisih, oratau perkalian productbilangan ofbulat integersbiasa.<ref>{{cite book|title=Algebra in Action: A Course in Groups, Rings, and Fields|volume=27|series=Pure and Applied Undergraduate Texts|first=Shahriar|last=Shahriari|author-link= Shahriar Shahriari |publisher=American Mathematical Society|year=2017|isbn=978-1-4704-2849-5|pages=20–21|url=https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA20}}</ref> EqualityKesamaan ofbilangan integersbulat correspondssesuai todengan '' congruencekongruensi'' indalam aritmetika modular arithmetic: ▼Modular sums, differences and products are calculated by performing the same replacement by the remainder
<math>x</math> dan <math>y</math> adalah kongruen (ditulis <math>x\equiv y</math> mod <math>n</math>) ketika mereka memiliki sisa yang sama setelah dibagi dengan <math>n</math>.<ref>{{harvnb|Dudley|1978}}, [https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA28 Teorema 3, hal. 28].</ref> Namun, dalam sistem bilangan ini, [[Pembagian (matematika)|pembagian]] dengan semua bilangan bukan nol dimungkinkan jika dan hanya jika modulusnya adalah prima. Misalnya, dengan bilangan prima <math>7</math> sebagai modulus, pembagian dengan <math>3</math> adalah dimungkinkan: <math>2/3\equiv 3\bmod{7}</math> karena kemungkinan [[menghapus penyebut]] dengan mengalikan kedua ruas dengan <math>3</math> diberikan rumus yang valid <math>2\equiv 9\bmod{7}</math>. Namun, dengan modulus komposit <math>6</math>, pembagian dengan <math>3</math> adalah hal mustahil. Tidak ada solusi yang valid untuk <math>2/3\equiv x\bmod{6}</math>: menghapus penyebut dengan mengalikan dengan <math>3</math> menyebabkan ruas kiri menjadi <math>2</math> sedangkan ruas kanan menjadi <math>0</math> atau <math>3 </math>. Dalam terminologi [[aljabar abstrak]], kemampuan untuk melakukan pembagian berarti bahwa modulo aritmatika modular bilangan prima membentuk [[medan (matematika)|medan]] atau [[medan berhingga]], sedangkan modulus lainnya hanya memberikan [[gelanggang (matematika)|gelanggang]] tetapi bukan sebuah medan.<ref>{{harvnb|Shahriari|2017}}, [https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA27 hal. 27–28].</ref>
▲on the result of the usual sum, difference, or product of integers.<ref>{{cite book|title=Algebra in Action: A Course in Groups, Rings, and Fields|volume=27|series=Pure and Applied Undergraduate Texts|first=Shahriar|last=Shahriari|author-link= Shahriar Shahriari |publisher=American Mathematical Society|year=2017|isbn=978-1-4704-2849-5|pages=20–21|url=https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA20}}</ref> Equality of integers corresponds to ''congruence'' in modular arithmetic:
<math>x</math> and <math>y</math> are congruent (written <math>x\equiv y</math> mod <math>n</math>) when they have the same remainder after division by <math>n</math>.<ref>{{harvnb|Dudley|1978}}, [https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA28 Theorem 3, p. 28].</ref> However, in this system of numbers, [[Division (mathematics)|division]] by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number <math>7</math> as modulus, division by <math>3</math> is possible: <math>2/3\equiv 3\bmod{7}</math>, because [[clearing denominators]] by multiplying both sides by <math>3</math> gives the valid formula <math>2\equiv 9\bmod{7}</math>. However, with the composite modulus <math>6</math>, division by <math>3</math> is impossible. There is no valid solution to <math>2/3\equiv x\bmod{6}</math>: clearing denominators by multiplying by <math>3</math> causes the left-hand side to become <math>2</math> while the right-hand side becomes either <math>0</math> or <math>3</math>.
In the terminology of [[abstract algebra]], the ability to perform division means that modular arithmetic modulo a prime number forms a [[field (mathematics)|field]] or, more specifically, a [[finite field]], while other moduli only give a [[ring (mathematics)|ring]] but not a field.<ref>{{harvnb|Shahriari|2017}}, [https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA27 pp. 27–28].</ref>
Beberapa teorema tentang bilangan prima dirumuskan menggunakan aritmetika modular. Misalnya, [[teorema kecil Fermat]] menyatakan bahwa jika <math>a\not\equiv 0</math> (mod <math>p</math>), thenmaka <math>a^{p-1}\equiv 1</math> (mod <math>p</math>).<ref>{{harvnb|Ribenboim|2004}}, Fermat'sTeorema littlekecil theoremFermat anddan primitiveakar rootsprimitif modulo a primeprima, pphal. 17–21.</ref> ▼
Several theorems about primes can be formulated using modular arithmetic. For instance, [[Fermat's little theorem]] states that if
Menjumlahkan dari semua pilihan <math>a</math> diberikan persamaan
▲<math>a\not\equiv 0</math> (mod <math>p</math>), then <math>a^{p-1}\equiv 1</math> (mod <math>p</math>).<ref>{{harvnb|Ribenboim|2004}}, Fermat's little theorem and primitive roots modulo a prime, pp. 17–21.</ref>
Summing this over all choices of <math>a</math> gives the equation
:<math>\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p,</math>
valid wheneverjika <math>p</math> isadalah primebilangan prima.
[[Giuga'sKonjektur conjectureGiuga]] saysmenyebutkan thatbahwa thispersamaan equationini isjuga alsomerupakan asyarat sufficientyang conditioncukup foruntuk <math>p</math> to bemenjadi primeprima.<ref>{{harvnb|Ribenboim|2004}}, The property of Giuga, pphal. 21–22.</ref>
[[Teorema Wilson's theorem]] saysmenyebutkan bahwa thatsebuah anbilangan integerbulat <math>p>1</math> isadalah primebilangan ifprima andjika onlydan ifhanya thejika [[factorialfaktorial]] <math>(p-1)!</math> iskongruen congruent todengan <math>-1</math> mod <math>p</math>. For a compositeUntuk {{nowrap|numberbilangan <math>\;n = r\cdot s\; </math>}} thisini cannottidak holdberlaku, sincekarena onesalah of its factorssatu dividesfaktornya bothmembagi {{mvar|n}} anddan <math>(n-1)!</math>, anddan sojadi <math>(n-1)!\equiv -1 \pmod{n}</math> isadalah impossiblehal mustahil.<ref>{{harvnb|Ribenboim|2004}}, The theorem of Wilson, phal. 21.</ref>
===Bilangan ''p''-adic numbersadik===
{{main|p-adicbilangan numberP-adik}}
[[urutan P-adik|Urutan <math>p</math>-adik]] <math>\nu_p(n)</math> dari sebuah bilangan bulat <math>n</math> adalah jumlah salinan dari <math>p</math> dalam faktorisasi prima dari <math>n</math>. Konsep yang sama diperluas dari bilangan bulat ke bilangan rasional dengan mendefinisikan urutan <math>p</math>-adik dari pecahan <math>m/n</math> menjadi <math>\nu_p(m)-\nu_p(n)</math>. Nilai absolut <math>p</math>-adik <math>|q|_p</math> dari sembarang bilangan rasional <math>q</math> kemudian didefinisikan sebagai <math>|q|_p=p^{-\nu_p(q)}</math>. Mengalikan bilangan bulat dengan nilai absolut <math>p</math>-adik-nya akan membatalkan faktor <math>p</math> dalam faktorisasinya, dan hanya menyisakan bilangan prima lainnya. Sama seperti jarak antara dua bilangan real yang dapat diukur dengan nilai absolut jaraknya, jarak antara dua bilangan rasional dapat diukur dengan jarak <math>p</math>-adik-nya, nilai absolut <math>p</math>-adik dari selisihnya. Untuk definisi jarak ini, dua bilangan dikatakan berdekatan (memiliki jarak yang kecil) ketika selisihnya habis dibagi dengan pangkat <math>p</math> yang tinggi. Dengan cara yang sama bahwa bilangan real dapat dibentuk dari bilangan rasional dan jaraknya, dengan menambahkan nilai pembatas ekstra untuk membentuk [[medan lengkap]], bilangan rasional dengan jarak <math>p</math>-adik diperluas ke medan lengkap yang berbeda<!--[[bilangan P-adik|bilangan <math>p</math>-adik]]-->.<ref name="childress"/><ref>{{cite book | last1 = Erickson | first1 = Marty | last2 = Vazzana | first2 = Anthony | last3 = Garth | first3 = David | edition = 2nd | isbn = 978-1-4987-1749-6 | mr = 3468748 | page = 200 | publisher = CRC Press | location = Boca Raton, FL | series = Textbooks in Mathematics | title = Introduction to Number Theory | url = https://books.google.com/books?id=QpLwCgAAQBAJ&pg=PA200 | year = 2016}}</ref>
The [[p-adic order|<math>p</math>-adic order]] <math>\nu_p(n)</math> of an integer <math>n</math> is the number of copies of <math>p</math> in the prime factorization of <math>n</math>. The same concept can be extended from integers to rational numbers by defining the <math>p</math>-adic order of a fraction <math>m/n</math> to be <math>\nu_p(m)-\nu_p(n)</math>. The <math>p</math>-adic absolute value <math>|q|_p</math> of any rational number <math>q</math> is then defined as
<math>|q|_p=p^{-\nu_p(q)}</math>. Multiplying an integer by its <math>p</math>-adic absolute value cancels out the factors of <math>p</math> in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their <math>p</math>-adic distance, the <math>p</math>-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of <math>p</math>. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a [[complete field]], the rational numbers with the <math>p</math>-adic distance can be extended to a different complete field, the [[p-adic number|<math>p</math>-adic numbers]].<ref name="childress"/><ref>{{cite book | last1 = Erickson | first1 = Marty | last2 = Vazzana | first2 = Anthony | last3 = Garth | first3 = David | edition = 2nd | isbn = 978-1-4987-1749-6 | mr = 3468748 | page = 200 | publisher = CRC Press | location = Boca Raton, FL | series = Textbooks in Mathematics | title = Introduction to Number Theory | url = https://books.google.com/books?id=QpLwCgAAQBAJ&pg=PA200 | year = 2016}}</ref>
ThisUrutan picturedari ofsebuah an ordergambar, absolutenilai valueabsolut, anddan completemedan fieldlengkap derivedyang fromditurunkan themdari canbilangan be<math>p</math>-adik generalizeddigeneralisasikan toke [[algebraicmedan numberbilangan fieldaljabar]]s and theirdan [[ValuationPenilaian (algebraaljabar)|valuationspenilaian-penilaian]] tersebut (certainpemetaan mappingstertentu fromdari theMedan [[multiplicativegrup groupperkalian]] of the field to ake [[totallygrup orderedterurut grouptotal|totallygrup orderedaditif additiveterurut grouptotal]], alsodisebut juga calledsebagai ordersurutan), [[AbsoluteNilai valueabsolut (algebraaljabar)|absolutenilai valuesabsolut]] (certainpemetaan multiplicativeperkalian mappingstertentu fromdari themedan fieldke to thebilangan real numbers,disebut alsojuga calledsebagai normsnorma),<ref name="childress">{{cite book | last = Childress | first = Nancy | doi = 10.1007/978-0-387-72490-4 | isbn = 978-0-387-72489-8 | mr = 2462595 | pages = 8–11 | publisher = Springer, New York | series = Universitext | title = Class Field Theory | url = https://books.google.com/books?id=RYdy4PCJYosC&pg=PA8 | year = 2009 }} SeeLihat alsopula phal. 64.</ref> anddan placestempat (extensionsekstensi toke [[completemedan fieldlengkap]]s indimana whichmedan theyang givendiberikan field is aadalah [[densehimpunan setrapat]], alsodisebut juga calledsebagai completionspelengkapan).<ref>{{cite book | last = Weil | first = André | author-link = André Weil | isbn = 978-3-540-58655-5 | mr = 1344916 | page = [https://archive.org/details/basicnumbertheor00weil_866/page/n56 43] | publisher = Springer-Verlag | location = Berlin | series = Classics in Mathematics | title = Basic Number Theory | url = https://archive.org/details/basicnumbertheor00weil_866 | url-access = limited | year = 1995}} NoteNamun howeverperhatikan thatbahwa somebeberapa authorspenulis such asseperti {{harvtxt|Childress|2009}} insteadmalah usemenggunakan "placetempat" tountuk meanmengartikan ankelas equivalencenorma classyang of normssetara.</ref> ThePerluasan extensiondari frombilangan therasional rational numbers to theke [[bilangan real number]]s, formisalnya instance,adalah istempat adimana placejarak in which the distance between numbers isantara thebilangan usualadalah [[absolutenilai valueabsolut]] ofbiasa theirdari differenceperbedaannya. ThePemetaan correspondingyang mappingsesuai toke angrup additiveaditif groupakan would be themenjadi [[logarithmlogaritma]] ofdari thenilai absolute valueabsolut, althoughmeskipun thisini doestidak notmemenuhi meetsemua allpersyaratan the requirements of a valuationpenilaian. According toMenurut [[teorema Ostrowski's theorem]], up to a naturalgagasan notionekuivalen ofalami equivalenceberhingga, thebilangan real numbersdan andbilangan <math>p</math>-adicadik numbers,dengan withurutan theirdan ordersnilai andabsolutnya absoluteadalah values, are the onlysatu-satunya valuationspenilaian, absolutenilai valuesabsolut, anddan placestempat onpada thebilangan rational numbersrasional.<ref name="childress"/> The [[localPrinsip lokal-global principle]] allowsmemungkinkan certainmasalah problemstertentu overatas thebilangan rationalrasional numbersuntuk todiselesaikan bedengan solvedmenyatukan bysolusi piecingdari together solutions from each of theirmasing-masing placestempat, againsekali underlininglagi themenggarisbawahi importancepentingnya ofbilangan primesprima tountuk numberteori theorybilangan.<ref>{{cite book | last = Koch | first = H. | doi = 10.1007/978-3-642-58095-6 | isbn = 978-3-540-63003-6 | mr = 1474965 | page = 136 | publisher = Springer-Verlag | location = Berlin | title = Algebraic Number Theory | url = https://books.google.com/books?id=wt1sCQAAQBAJ&pg=PA136 | year = 1997| citeseerx = 10.1.1.309.8812 }}</ref>
===Prime elements in rings===
which states that an odd prime <math>p</math> is expressible as the sum of two squares, <math>p=x^2+y^2</math>, and therefore factorizable as <math>p=(x+iy)(x-iy)</math>, exactly when <math>p</math> is 1 mod 4.<ref>{{harvnb|Kraft|Washington|2014}}, [https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA297 Section 12.1, Sums of two squares, pp. 297–301].</ref>
===Prime idealsIdeal prima ===
{{Main|PrimeIdeal idealsprima}}
NotTidak everysemua ringgelanggang ismerupakan aranah uniquefaktorisasi factorization domainunik. For instanceMisalnya, in the ringdalam ofbilangan numbersgelanggang <math>a+b\sqrt{-5}</math> (foruntuk integersbilangan bulat <math>a</math> anddan <math>b</math>) the numberangka <math>21</math> hasmemiliki twodua factorizationsfaktorisasi <math>21=3\cdot7=(1+2\sqrt{-5})(1-2\sqrt{-5})</math>, wheretidak neithersatu ofpun thedari fourkeempat factorsfaktor cantersebut bebisa reduceddireduksi anylebih furtherjauh, sosehingga ittidak doesmemiliki notfaktorisasi have a unique factorizationunik. InUntuk ordermemperluas tofaktorisasi extendunik uniquepada factorizationkelas togelanggang a larger class of ringsterbesar, the notion of a number can begagasan replacedtentang withbilangan thatbisa ofdiganti andengan [[ideal (ringteori theorygelanggang)|ideal]], asebuah subsethimpunan ofbagian thedari elementselemen ofgelanggang ayang ringmemuat thatsemua containsjumlah allpasangan sums of pairs of its elementselemennya, and alldan productssemua ofhasil itskali elementselemennya withdengan ringelemen elementsgelanggang.
''PrimeIdeal idealsprima'', whichyang generalizedimana primegeneralisasi elementselemen inprima thedalam sensearti that thebahwa [[principalideal idealutama]] generatedyang bydihasilkan aoleh primeelemen elementprima is a primeadalah ideal, are anprima importantadalah toolalat anddan objectobjek ofstudi studypenting indalam [[commutativealjabar algebrakomutatif]], [[numberteori theorybilangan|algebraicteori numberbilangan theoryaljabar]] anddan [[algebraicgeometri geometryaljabar]]. TheIdeal primeprima idealsdari ofgelanggang thebilangan ringbulat ofadalah integers are the idealsideal (0), (2), (3), (5), (7), (11), ... TheTeorema fundamentaldasar theoremaritmetika ofdigeneralisasikan arithmetic generalizes to theke [[teorema Lasker–Noether theorem]], whichdisebutkan expresses everysetiap ideal in adalam [[Noetheriangelanggang ring|Noetheriankomutatif]] [[commutativegelanggang ringNoetherian|Noetherian]] assebagai an intersection ofperpotongan [[primary ideal prima]]s, whichyang aremerupakan thegeneralisasi appropriateyang generalizationstepat ofdari [[primeprima powerkuasa]]s.<ref>{{cite book | last1=Eisenbud | first1=David | author1-link= David Eisenbud | title=Commutative Algebra | publisher=Springer-Verlag | location=Berlin; New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1| mr=1322960 | year=1995 | volume=150 | at=Section 3.3| doi=10.1007/978-1-4612-5350-1 }}</ref>
The [[spectrumSpektrum of a ringgelanggang]] isadalah aruang geometricgeometris spaceyang whosetitik-titiknya pointsmerupakan areideal theprima prime ideals ofdari thegelanggang ringtersebut.<ref>{{cite book | last = Shafarevich | first = Igor R. | author-link = Igor Shafarevich | doi = 10.1007/978-3-642-38010-5 | edition = 3rd | isbn = 978-3-642-38009-9 | mr = 3100288 | publisher = Springer, Heidelberg | title = Basic Algebraic Geometry 2: Schemes and Complex Manifolds | year = 2013 | contribution = Definition of <math>\operatorname{Spec} A</math> | page = 5 | contribution-url = https://books.google.com/books?id=zDW8BAAAQBAJ&pg=PA5}}</ref> [[ArithmeticGeometri geometryaritmetika]] alsojuga benefitsmendapat frommanfaat thisdari notiongagasan ini, anddan manybanyak conceptskonsep existyang inada, bothbaik geometrydalam andgeometri numbermaupun theoryteori bilangan. For exampleMisalnya, factorizationfaktorisasi oratau [[SplittingPemisah ofideal primeprima idealsdalam inperluasan Galois extensions|ramificationpercabangan]] ofdari primeideal idealsprima whenketika lifteddiangkat to ansebagai [[field extension|extensionmedan fieldperluasan]], amasalah basicdasar problemteori ofbilangan algebraicaljabar numbermemiliki theory, bears somebeberapa resemblancekemiripan withdengan [[ramifiedpeliput coverbercabang|ramificationpercabangan indalam geometrygeometri]]. TheseKonsep-konsep conceptsini canbahkan evendapat assistmembantu withdalam inpertanyaan number-theoreticteori questionsbilangan solelyyang concernedhanya withberkaitan integers.dengan Forbilangan bulat. exampleMisalnya, primeideal idealsprima in thedalam [[ringgelanggang ofbilangan integersbulat]] ofdari [[quadraticmedan numberbilangan fieldkuadrat]]s candapat bedigunakan useduntuk in provingpenggunaan [[quadraticketimbalbalikan reciprocitykuadrat]], apernyataan statementyang thatmenyangkut concernskeberadaan theakar existencekuadrat ofmodulo square roots modulobilangan integerprima primebilangan numbersbulat.<ref>{{cite book | last = Neukirch | first = Jürgen | author-link = Jürgen Neukirch | doi = 10.1007/978-3-662-03983-0 | isbn = 978-3-540-65399-8 | location = Berlin | mr = 1697859 | publisher = Springer-Verlag | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | title = Algebraic Number Theory | volume = 322 | year = 1999 | at = Section I.8, phal. 50}}</ref>
EarlyUpaya attemptsawal tountuk provemembuktikan [[Fermat'sTeorema LastTerakhir TheoremFermat]] ledmenyebabkan topengenalan [[Ernst Kummer|Kummer]]'s introduction ofdari [[regularprima primeregular]]s, integerbilangan primeprima numbersbilangan connectedbulat withterhubung thedengan failurekegagalan offaktorisasi uniqueunik factorization in thepada [[cyclotomicmedan fieldsiklotomi|cyclotomicbilangan bulat integerssiklotomi]].<ref>{{harvnb|Neukirch|1999}}, SectionBagian I.7, phal. 38</ref>
ThePertanyaan questiontentang ofberapa howbanyak manybilangan integerprima primebilangan numbersbulat factorfaktor intomenjadi adarab productdari ofbeberapa multipleideal primeprima idealsdalam inmedan anbilangan algebraicaljabar numberditangani field is addressed byoleh [[Chebotarev'steorema densitykerapatan theoremChebotarev]], whichyang (whenbila appliedditerapkan topada thebilangan cyclotomicbulat integerssiklotomi) hasmana Dirichlet'smemiliki teorema theoremDirichlet onpada primesbilangan inprima arithmeticdalam progressionsderet asaritmatika asebagai specialkasus casekhusus.<ref>{{cite journal | last1 = Stevenhagen | first1 = P. | last2 = Lenstra | first2 = H.W., Jr. | author2-link = Hendrik Lenstra | doi = 10.1007/BF03027290 | issue = 2 | journal = [[The Mathematical Intelligencer]] | mr = 1395088 | pages = 26–37 | title = Chebotarëv and his density theorem | volume = 18 | year = 1996| citeseerx = 10.1.1.116.9409 | s2cid = 14089091 }}</ref>
===Group theory===
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