U [[number theory|teoriji brojeva]], '''teorema prostih brojeva''' ({{jez-eng-lat|prime number theorem}}, '''PNT''') opisuje [[asymptotic analysis|asimptotsku]] distribuciju [[prime number|prostih brojeva]] među positivnim celim brojevima. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by [[Jacques Hadamard]] and [[Charles Jean de la Vallée Poussin]] in 1896 using ideas introduced by [[Bernhard Riemann]] (in particular, the [[Riemann zeta function]]).▼
▲U [[number theory|teoriji brojeva]], '''teorema prostih brojeva''' ({{jez-eng-lat|prime number theorem}}, -{'''PNT'''}-) opisuje [[asymptotic analysis|asimptotsku]] distribuciju [[prime number|prostih brojeva]] među positivnimpozitivnim celim brojevima. ItTo formalizesformalizuje theintuitivnu intuitiveideju ideada thatprosti primesbrojevi becomepostaju lessmanje commonzastupljeni askako theypostaju becomeveći largeru byskladu preciselysa quantifyingprecizno thekvantifikovanom ratestopom atkojom whichdo thistoga occursdolazi. Teoremu Thesu theoremnezavisno was proved independently bydokazali [[JacquesŽak HadamardAdamar]] andi [[Charles Jean de la Vallée Poussin|Šarl Žan de la Vale-Pusen]] in 1896. usinggodine, ideaskoristeći introducedideje bykoje je uveo [[Bernhard RiemannRiman]] (in particular, thenaročito [[RiemannРиманова зета-функција|Rimanovu zeta functionfunkciju]]).
The first such distribution found is {{math|''π''(''N'') ~ {{sfrac|''N''|log(''N'')}}}}, where {{math|''π''(''N'')}} is the [[prime-counting function]] and {{math|log(''N'')}} is the [[natural logarithm]] of {{mvar|N}}. This means that for large enough {{mvar|N}}, the [[probability]] that a random integer not greater than {{mvar|N}} is prime is very close to {{math|1 / log(''N'')}}. Consequently, a random integer with at most {{math|2''n''}} digits (for large enough {{mvar|n}}) is about half as likely to be prime as a random integer with at most {{mvar|n}} digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime ({{math|log(10<sup>1000</sup>) ≈ 2302.6}}), whereas among positive integers of at most 2000 digits, about one in 4600 is prime ({{math|log(10<sup>2000</sup>) ≈ 4605.2}}). In other words, the average gap between consecutive prime numbers among the first {{mvar|N}} integers is roughly {{math|log(''N'')}}.<ref>{{cite book|last = Hoffman|first = Paul|title = The Man Who Loved Only Numbers|url = https://archive.org/details/manwholovedonlyn00hoff/page/227|url-access = registration|publisher = Hyperion Books|year = 1998|page = [https://archive.org/details/manwholovedonlyn00hoff/page/227 227]|isbn = 978-0-7868-8406-3|mr = 1666054|location = New York}}</ref>▼
▲ThePrva firsttakva suchraspodela distribution found isje {{math|''π''(''N'') ~ {{sfrac|''N''|log(''N'')}}}}, wheregde je {{math|''π''(''N'')}} is the [[prime-countingfunkcija raspodele prostih functionbrojeva]] andi {{math|log(''N'')}} is theje [[naturalprirodni logarithmlogaritam]] ofod {{mvar|N}}. ThisTo meansznači thatda forza largedovoljno enoughveliko {{mvar|N}}, the [[probabilityverovatnoća]] thatda aje randomslučajni integerceli notbroj greaterkoji thannije veći od {{mvar|N}} isprost primebroj isvrlo very close toblizu {{math|1 / log(''N'')}}. ConsequentlySledstveno tome, aza randomslučajni integerceli withbroj atsa mostnajviše {{math|2''n''}} digitscifara (forza largedovoljno enoughveliko {{mvar|n}}) ispostoji aboutaproksimativno halfupola asmanja likelyverovatnoća toda beće primeon asbiti aprost randombroj integerkao withslučajni atceli mostbroj sa najviše {{mvar|n}} digitscifara. ForNa exampleprimer, amongmeđu thepozitivnim positivecelim integersbrojevima ofod at mostnajviše 1000 digitscifara, aboutjedan one inod 2300 isje prost primebroj ({{math|log(10<sup>1000</sup>) ≈ 2302.,6}}), whereasdok je amongmeđu positivepozitivnim integerscelim ofbrojevima atsa mostnajviše 2000 digitscifara, aboutpribližno onejedan inod 4600 isprost primebroj ({{math|log(10<sup>2000</sup>) ≈ 4605.,2}}). InDrugim other wordsrečima, theprosečni averagerazmak gapizmeđu betweenuzastopnih consecutiveprostih primebrojeva numbersmeđu among the firstprvih {{mvar|N}} integerscelih brojevima isje roughlyoko {{math|log(''N'')}}.<ref>{{cite book|last = Hoffman|first = Paul|title = The Man Who Loved Only Numbers|url = https://archive.org/details/manwholovedonlyn00hoff/page/227|url-access = registration|publisher = Hyperion Books|year = 1998|page = [https://archive.org/details/manwholovedonlyn00hoff/page/227 227]|isbn = 978-0-7868-8406-3|mr = 1666054|location = New York}}</ref>
