Ir para o conteúdo
Number Buffet
pt

Os primeiros 30 primos de mersenne

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151, 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127, 10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087, 1475979915214180235084898622737381736312066145333169775147771216478570297878078949377407337049389289382748507531496480477281264838760259191814463365330269540496961201113430156902396093989090226259326935025281409614983499388222831448598601834318536230923772641390209490231836446899608210795482963763094236630945410832793769905399982457186322944729636418890623372171723742105636440368218459649632948538696905872650486914434637457507280441823676813517852099348660847172579408422316678097670224011990280170474894487426924742108823536808485072502240519452587542875349976558572670229633962575212637477897785501552646522609988869914013540483809865681250419497686697771007, 446087557183758429571151706402101809886208632412859901111991219963404685792820473369112545269003989026153245931124316702395758705693679364790903497461147071065254193353938124978226307947312410798874869040070279328428810311754844108094878252494866760969586998128982645877596028979171536962503068429617331702184750324583009171832104916050157628886606372145501702225925125224076829605427173573964812995250569412480720738476855293681666712844831190877620606786663862190240118570736831901886479225810414714078935386562497968178729127629594924411960961386713946279899275006954917139758796061223803393537381034666494402951052059047968693255388647930440925104186817009640171764133172418132836351, 259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071, 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34608828249085121524296039576741331672262866890023854779048928344500622080983411446436437554415370753366448674763505018641470709332373970608376690404229265789647993709760358469552319045484910050304149809818540283507159683562232941968059762281334544739720849260904855192770626054911793590389060795981163838721432994278763633095377438194844866471124967685798888172212033000821469684464956146997194126921284336206463313859537577200462442029064681326087558257488470489384243989270236884978643063093004422939603370010546595386302009073043944482202559097406700597330570799507832963130938739885080198416258635194522913042562936679859587495721031173747796418895060701941717506001937152430032363631934265798516236047451209089864707430780362298307038193445486493756647991804258775574973833903315735082891029392359352758617185019942554834671861074548772439880729606244911940066680112823824095816458261761861746604034802056466823143718255492784779380991749580255263323326536457743894150848953969902818530057870876229329803338285735419228259022169602665532210834789602051686546011466737981306056247480055071718250333737502267307344178512950738594330684340802698228963986562732597175372087295649072830289749771358330867951508710859216743218522918811670637448496498549094430541277444079407989539857469452772132166580885754360477408842913327292948696897496141614919739845432835894324473601387609643750514699215032683744527071718684091832170948369396280061184593746143589068811190253101873595319156107319196071150598488070027088705842749605203063194191166922106176157609367241948160625989032127984748081075324382632093913796444665700601391278360323002267434295194325607280661260119378719405151497555187549252134264394645963853964913309697776533329401822158003182889278072368602128982710306618115118964131893657845400296860012420391376964670183983594954112484565597312460737798777092071706710824503707457220155015899591766244957768006802482976673920392995410164224776445671222149803657927708412925555542817045572430846389988129960519227313987291200902060882060733762075892299473666405897427035811786879875694315078654420055603469625309399653955932310466430039146465805452965014040019423897552675534768248624631951431493188170905972588780111850281190559073677771187432814088678674286302108275149258477101296451833651979717375170900505673645964696355331369819296000267389583289299126738345726980325998955997501176664201042888546085699446442834195232948787488410595750197438786353119204210855804692460582533832967771946911459901921324984968810021189968284941331573164056304725480868921823442538199590383852412786840833479611419970101792978355653650755329138298654246225346827207503606740745956958127383748717825918527473164970582095181312905519242710280573023145554793628499010509296055849712377978984921839997037415897674154830708629145484724536724572622450131479992681684310464449439022250504859250834761894788889552527898400988196200014868575640233136509145628127191354858275083907891469979019426224883789463551, 2^11213 - 1 (3376 digits), 2^19937 - 1 (6002 digits), 2^21701 - 1 (6533 digits), 2^23209 - 1 (6987 digits), 2^44497 - 1 (13395 digits), 2^86243 - 1 (25962 digits), 2^110503 - 1 (33265 digits), 2^132049 - 1 (39751 digits)

8 of these are shown in power form rather than written out — past 2^9,941 - 1 the digits run into the thousands and then the millions.

Configurações

Predefinições rápidas

Only 52 Mersenne primes have ever been found, so 52 is the ceiling.

1 is 2^2 - 1 = 3. Set 48 to begin at the 48th known Mersenne prime.

Full decimal form switches to 2^p - 1 once a prime is too long to print.

Group digits as 2,147,483,647.

Ajustar a aparência

Escolha primeiro uma predefinição ao lado da imagem — estes controles a ajustam.

Frame

A border drawn inside the edge of the image.

Avançado

Resultados

30 valores

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151, 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127, 10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087, 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2^11213 - 1 (3376 digits), 2^19937 - 1 (6002 digits), 2^21701 - 1 (6533 digits), 2^23209 - 1 (6987 digits), 2^44497 - 1 (13395 digits), 2^86243 - 1 (25962 digits), 2^110503 - 1 (33265 digits), 2^132049 - 1 (39751 digits)

8 of these are shown in power form rather than written out — past 2^9,941 - 1 the digits run into the thousands and then the millions.

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Quais são os primeiros 30 primos de mersenne?

Os primeiros 30 primos de mersenne são:

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151, 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127, 10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087, 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2^11213 - 1 (3376 digits), 2^19937 - 1 (6002 digits), 2^21701 - 1 (6533 digits), 2^23209 - 1 (6987 digits), 2^44497 - 1 (13395 digits), 2^86243 - 1 (25962 digits), 2^110503 - 1 (33265 digits), 2^132049 - 1 (39751 digits)

O artigo de contexto abaixo ainda não foi traduzido e aparece em inglês.

Sobre primos de mersenne

Marin Mersenne was a French Minim friar, music theorist and tireless correspondent who acted as a clearing house for European mathematics in the 1630s and 1640s. In Cogitata Physico-Mathematica (1644) he asserted that 2^p - 1 is prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and composite for every other p below 257. He gave no proof, and he got it wrong in five places: 67 and 257 are composite, and he left out 61, 89 and 107. It took nearly three centuries to finish checking a single sentence.

Euler settled 2^31 - 1 in 1772. In 1876 Édouard Lucas proved 2^127 - 1 prime — thirty-nine digits, by hand, and still the largest prime ever found without a machine — and in the same work showed that Mersenne's 2^67 - 1 was composite without producing a factor. That loose end became one of mathematics' better anecdotes: at an American Mathematical Society meeting in New York in 1903, Frank Nelson Cole walked to the blackboard, silently worked out 2^67 - 1, then multiplied 193,707,721 by 761,838,257,287 to get the same number, and sat down to applause without having spoken. He later said the calculation had taken him "three years of Sundays".

Machines took over in 1952, when Raphael Robinson ran Lucas's test — sharpened into the Lucas–Lehmer test by Derrick Henry Lehmer around 1930 — on the SWAC computer in Los Angeles and found five new Mersenne primes in a single year. In 1996 George Woltman launched the Great Internet Mersenne Prime Search, and every Mersenne prime from the 35th onwards has come out of it. The 52nd and largest, 2^136,279,841 - 1, was reported in October 2024 by Luke Durant using rented GPU capacity: 41,024,320 digits.

Propriedades principais

  • 2^n - 1 can only be prime when n is itself prime, because 2^ab - 1 is always divisible by 2^a - 1.
  • The converse fails: 11 is prime but 2^11 - 1 = 2047 = 23 × 89.
  • 52 Mersenne primes are known. Every exponent below the 48th (57,885,161) has been tested, so the first 48 are consecutive; gaps may remain between the larger ones.
  • Euclid showed that 2^(p-1)(2^p - 1) is perfect whenever 2^p - 1 is prime; Euler proved the converse, so even perfect numbers and Mersenne primes are in exact correspondence.
  • The Lucas–Lehmer test decides primality of 2^p - 1 for odd prime p by iterating s := s² - 2 modulo 2^p - 1, starting from s = 4; the number is prime exactly when the (p-2)th value is 0.
  • If p is a prime congruent to 3 modulo 4 and 2p + 1 is also prime, then 2p + 1 divides 2^p - 1 — which is how 23 divides 2047.
  • In binary a Mersenne prime is a string of p ones, making these the base-2 repunit primes.
  • Nobody knows whether infinitely many Mersenne primes exist, nor whether infinitely many Mersenne numbers with prime exponent are composite.

Outras quantidades

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