梅森数及梅森素因子

yangchuanju · 发表于 2022-10-17 04:46

梅森数各个素因子和复合因子之间的关系

除2^2-1=3以外，所有的梅森数2^p-1都是2kp+1型的整数：
2^3-1=7=2*1*3+1,k=1;
2^5-1=31=2*3*5+1,k=3;
2^7-1=127=2*9*7+1,k=9;
2^11-1=2047=2*93*11+1,k=93;……

梅森数的素因子都是2kp+1型的整数：
2^11-1=2047=23*89,23=2*1*11+1,k=1;89=2*4*11+1,k=4;
2^23-1=8388607=47*178481,47=2*1*23+1,k=1;178481=2*3880*23+1,k=3880;
2^29-1=536870911=233*1103*2089,233=2*4*29+1,k=4;1103=2*19*29+1,k=19;2089=2*36*29+1,k=36;……

梅森数的复合因子也是2kp+1型的整数：
2^29-1=536870911=233*1103*2089,233*1103=256999=2*4431*29+1,k=4431;
233*2089=486737=2*8392*29+1,k=8392;1103*2089=2304167=2*39727*29+1,k=39727。
2^43-1=8796093022207=431*9719*2098863,最小素因子431之k=5，其它素因子和复合因子之k分别等于113,24417,48708,10523732,237308936，都是整数。

多因子梅森数的各个素因子的k或复合因子的k之间一般不能整除，但也有些能整除：
梅森数2^11-1、2^23-1之小素因子对应的k等于1，大素因子的k肯定能整除小因子的k；
梅森数2^29-1的3个素因子对应的k分别等于4、19、36，36可以整除4，但19不能整除4，36也不能整除19。

yangchuanju · 发表于 2022-10-17 04:47

本帖最后由 yangchuanju 于 2022-10-17 05:06 编辑

经统计，在前1229个指数n小于10000的梅森数中，共有22个梅森素数，其余都是合数；
在那些合数之中，有280个被完全分解，素因子个数2-9个；
有873个尚未完全分解，最小的是2^1213-1、2^1217-1、2^1229-1、2^1231-1；
有54个至今尚未找到任何素因子的，最小的是2^1277-1。

yangchuanju · 发表于 2022-10-17 04:50

前22个梅森素数之p值及位数：
p 位数
2 1
3 1
5 2
7 3
13 4
17 6
19 6
31 10
61 19
89 27
107 33
127 39
521 157
607 183
1279 386
2203 664
2281 687
3217 969
4253 1281
4423 1332
9689 2917
9941 2993

yangchuanju · 发表于 2022-10-17 05:00

最小的已被完全分解的280个梅森数（按因子个数排序）：
p 位数因子数分解式
11 4 2 2047 = 23×89
23 7 2 8388607 = 47×178481
37 12 2 137438953471<12> = 223×616318177
41 13 2 2199023255551<13> = 13367×164511353
59 18 2 576460752303423487<18> = 179951×3203431780337<13>
67 21 2 147573952589676412927<21> = 193707721×761838257287<12>
83 25 2 9671406556917033397649407<25> = 167×57912614113275649087721<23>
97 30 2 158456325028528675187087900671<30> = 11447×13842607235828485645766393<26>
101 31 2 2535301200456458802993406410751<31> = 7432339208719<13>×341117531003194129<18>
103 32 2 10141204801825835211973625643007<32> = 2550183799<10>×3976656429941438590393<22>
109 33 2 649037107316853453566312041152511<33> = 745988807×870035986098720987332873<24>
131 40 2 2722258935367507707706996859454145691647<40> = 263×10350794431055162386718619237468234569<38>
137 42 2 174224571863520493293247799005065324265471<42> = 32032215596496435569<20>×5439042183600204290159<22>
139 42 2 696898287454081973172991196020261297061887<42> = 5625767248687<13>×123876132205208335762278423601<30>
149 45 2 713623846352979940529142984724747568191373311<45> = 86656268566282183151<20>×8235109336690846723986161<25>
167 51 2 2^167-1<51> = 2349023×79638304766856507377778616296087448490695649<44>
197 60 2 2^197-1<60> = 7487×2682880399...33<56>
199 60 2 2^199-1<60> = 164504919713<12>×4884164093883941177660049098586324302977543600799<49>
227 69 2 2^227-1<69> = 26986333437777017<17>×7992177738...31<52>
241 73 2 2^241-1<73> = 22000409×1606194743...39<66>
269 81 2 2^269-1<81> = 13822297×6862598850...63<74>
271 82 2 2^271-1<82> = 15242475217<11>×2489277578...91<72>
281 85 2 2^281-1<85> = 80929×4800921529...19<80>
293 89 2 2^293-1<89> = 40122362455616221971122353<26>×3966452270...47<63>
347 105 2 2^347-1<105> = 14143189112952632419639<23>×2027034530...93<83>
373 113 2 2^373-1<113> = 25569151×7524403464...41<105>
379 115 2 2^379-1<115> = 180818808679<12>×6809649408...53<103>
421 127 2 2^421-1<127> = 614002928307599<15>×8819779591...49<112>
457 138 2 2^457-1<138> = 150327409×2475539419...19<130>
487 147 2 2^487-1<147> = 4871×8203321996...37<143>
523 158 2 2^523-1<158> = 1601887783...63<69>×1714176918...89<90>
727 219 2 2^727-1<219> = 1760629171...27<98>×4009949972...01<122>
809 244 2 2^809-1<244> = 4148386731...37<61>×8229761617...03<183>
881 266 2 2^881-1<266> = 26431×6099757718...21<261>
971 293 2 2^971-1<293> = 2391710497...13<53>×8344823973...19<240>
983 296 2 2^983-1<296> = 1808226257914551209964473260866417929207023<43>×4520983959...09<254>
997 301 2 2^997-1<301> = 1675608165...07<57>×7993430605...53<244>
1061 320 2 2^1061-1<320> = 4681722635...33<143>×5277396428...47<177>
1063 320 2 2^1063-1<320> = 1485761479<10>×6651755792...33<311>
1427 430 2 2^1427-1<430> = 19054580564725546974193126830978590503<38>×1948966701...09<393>
1487 448 2 2^1487-1<448> = 24464753918382797416777<23>×1750099338...51<426>
1637 493 2 2^1637-1<493> = 81679753×7481496656...07<485>
1657 499 2 2^1657-1<499> = 1078842615...83<56>×5939429366...37<444>
2357 710 2 2^2357-1<710> = 6674719305...11<62>×5049713899...61<648>
2927 882 2 2^2927-1<882> = 1217183584262023230020873<25>×1070141538...99<858>
3079 927 2 2^3079-1<927> = 25324846649810648887383180721<29>×2936363555...47<899>
3259 982 2 2^3259-1<982> = 21926805872270062496819221124452121<35>×5197332446...47<947>
3359 1012 2 2^3359-1<1012> = 6719×2150061062...73<1008>
4111 1238 2 2^4111-1<1238> = 1081996892...31<56>×3162905096...37<1183>
4243 1278 2 2^4243-1<1278> = 101833×1829713380...79<1273>
4729 1424 2 2^4729-1<1424> = 61944189981415866671112479477273<32>×6009646362...07<1392>
5689 1713 2 2^5689-1<1713> = 919724609777<12>×3944459715...43<1701>
6043 1820 2 2^6043-1<1820> = 11155520642419038056369903183<29>×1193366805...29<1792>
6679 2011 2 2^6679-1<2011> = 3206486298...21<54>×1183890815...47<1958>
7331 2207 2 2^7331-1<2207> = 458072843161<12>×1548687051...27<2196>
7757 2336 2 2^7757-1<2336> = 233293220467553594643512097574361<33>×5269558706...11<2303>
29 9 3 536870911 = 233×1103×2089
43 13 3 8796093022207<13> = 431×9719×2099863
47 15 3 140737488355327<15> = 2351×4513×13264529
53 16 3 9007199254740991<16> = 6361×69431×20394401
71 22 3 2361183241434822606847<22> = 228479×48544121×212885833
73 22 3 9444732965739290427391<22> = 439×2298041×9361973132609<13>
79 24 3 604462909807314587353087<24> = 2687×202029703×1113491139767<13>
179 54 3 2^179-1<54> = 359×1433×1489459109360039866456940197095433721664951999121<49>
193 59 3 2^193-1<59> = 13821503×61654440233248340616559<23>×14732265321145317331353282383<29>
211 64 3 2^211-1<64> = 15193×60272956433838849161<20>×3593875704495823757388199894268773153439<40>
257 78 3 2^257-1<78> = 535006138814359<15>×1155685395246619182673033<25>×374550598501810936581776630096313181393<39>
277 84 3 2^277-1<84> = 1121297×31133636305610209482201109050392404721<38>×6955979459776540052280934851589652278783<40>
283 86 3 2^283-1<86> = 9623×68492481833<11>×2357954301...73<71>
311 94 3 2^311-1<94> = 5344847×2647649373910205158468946067671<31>×2948036813...31<57>
331 100 3 2^331-1<100> = 16937389168607<14>×865118802936559<15>×2985426249...19<72>
349 106 3 2^349-1<106> = 1779973928671<13>×34720396273212657799920861294559<32>×1855539364...99<62>
353 107 3 2^353-1<107> = 931921×2927455476800301964116805545194017<34>×6725414756...63<67>
389 118 3 2^389-1<118> = 56478911×4765678679<10>×4684435266...19<100>
409 124 3 2^409-1<124> = 4480666067023<13>×76025626689833<14>×3881196575...29<97>
443 134 3 2^443-1<134> = 887×207818990653657<15>×1232194392...73<117>
467 141 3 2^467-1<141> = 121606801×1148984537...47<58>×2727318928...41<76>
499 151 3 2^499-1<151> = 20959×1998447222711143545931606352264121<34>×3907550462...33<113>
563 170 3 2^563-1<170> = 2815747080256641401887817<25>×13299213974872825932452460424235657<35>×8062466865...03<111>
577 174 3 2^577-1<174> = 3463×132305774316967<15>×1079633141...51<157>
599 181 3 2^599-1<181> = 16659379034607403556537<23>×148296291984475077955727317447564721950969097<45>×8398047009...83<114>
613 185 3 2^613-1<185> = 44599476833089207<17>×332817722770314187794325446534549089<36>×2290082526...17<133>
631 190 3 2^631-1<190> = 333628015107245479<18>×474640860193534882628078580680807822523991<42>×5627294314...23<131>
643 194 3 2^643-1<194> = 3189281×2253242905...53<71>×5079095912...99<117>
647 195 3 2^647-1<195> = 303303806129303896428103<24>×11502383762931955619183430785983<32>×1673946143...23<141>
683 206 3 2^683-1<206> = 1367×4348364991...69<78>×6751379697...09<125>
709 214 3 2^709-1<214> = 216868921×1391646221255097953<19>×8923613716...47<187>
751 227 3 2^751-1<227> = 2276402451...87<66>×6493500319...97<67>×8013068084...73<94>
769 232 3 2^769-1<232> = 1591805393<10>×6123566623856435977170641<25>×3185460664...47<198>
829 250 3 2^829-1<250> = 72953×89654239955972104838901500177<29>×5473339866...31<216>
919 277 3 2^919-1<277> = 33554520197234177<17>×1526510740...21<126>×8651972246...11<135>
941 284 3 2^941-1<284> = 7529×2919837555...37<69>×8455317738...87<211>
1039 313 3 2^1039-1<313> = 5080711×5585366661...53<80>×2075818194...89<227>
1103 333 3 2^1103-1<333> = 2207×4126110275598714647074087<25>×1193279152...23<305>
1117 337 3 2^1117-1<337> = 53617×5209739697...09<78>×6373626673...07<254>
1123 339 3 2^1123-1<339> = 777288435261989969<18>×4321749374...87<86>×3391902395...69<235>
1171 353 3 2^1171-1<353> = 153606920351<12>×1234867096...13<128>×1690806550...69<215>
1193 360 3 2^1193-1<360> = 121687×8522732620...29<104>×1297065115...17<251>
1301 392 3 2^1301-1<392> = 161317830296866767945829203107381353<36>×3172016620920990460580749619964431858642380903<46>×8531129458...89<311>
1303 393 3 2^1303-1<393> = 10444849×1140690503<10>×1465596295...81<377>
1327 400 3 2^1327-1<400> = 2730967×6363071161<10>×1685858087...21<384>
1459 440 3 2^1459-1<440> = 14591×93377×1170681349...41<431>
1531 461 3 2^1531-1<461> = 88799×14661821742553<14>×5785319541...01<443>
1553 468 3 2^1553-1<468> = 14194270913<11>×354271470446427666439<21>×432076532964254217618937<24>×67625467049371226906709645391<29>×169336513704068271013494661928987033<36>×1269733052...83<350>
1559 470 3 2^1559-1<470> = 3119×1381449566...01<52>×4692592378...73<415>
1907 575 3 2^1907-1<575> = 1685696416143478298579153<25>×5495792295672019805118568131602993249446111<43>×1251388203...69<508>
2311 696 3 2^2311-1<696> = 77567729423209<14>×4514379640917651135021865565129<31>×1367857392...27<652>
2383 718 3 2^2383-1<718> = 1132387303<10>×889865441849036810633962436005361<33>×2244710483...29<676>
2887 870 3 2^2887-1<870> = 23135734040831<14>×8780552788...03<56>×5831661945...39<800>
3041 916 3 2^3041-1<916> = 24329×5565031×1998135647...49<905>
3547 1068 3 2^3547-1<1068> = 1948447035193<13>×148823192092809407<18>×1954498762...77<1039>
3833 1154 3 2^3833-1<1154> = 14193959303<11>×340789152474053904109001<24>×1456746763...97<1121>
4127 1243 3 2^4127-1<1243> = 74287×2080009×1451490076...69<1232>
4507 1357 3 2^4507-1<1357> = 21381209×1601788495...01<53>×1612697930...03<1298>
4871 1467 3 2^4871-1<1467> = 9743×144558074148074062894378117006327<33>×1473580354...27<1431>
6883 2072 3 2^6883-1<2072> = 1885943×2043031664890199<16>×2533124262...51<2051>
7673 2310 3 2^7673-1<2310> = 184153×13918823×2479564553...89<2298>
8233 2479 3 2^8233-1<2479> = 195611422708913289323830429299939553967<39>×20866399143857514113654051017725144974702596369<47>×5876416631...17<2394>
8849 2664 3 2^8849-1<2664> = 52368383×15264764469472455023<20>×8159665559...79<2637>
9697 2920 3 2^9697-1<2920> = 724126946527<12>×19092282046942032847<20>×8855139416...59<2888>

yangchuanju · 发表于 2022-10-17 05:01

最小的已被完全分解的280个梅森数（按因子个数排序）：
p 位数因子数分解式
157 48 4 182687704666362864775460604089535377456991567871<48> = 852133201×60726444167<11>×1654058017289<13>×2134387368610417<16>
173 53 4 2^173-1<53> = 730753×1505447×70084436712553223<17>×155285743288572277679887<24>
181 55 4 2^181-1<55> = 43441×1164193×7648337×7923871097285295625344647665764672671<37>
229 69 4 2^229-1<69> = 1504073×20492753×59833457464970183<17>×467795120187583723534280000348743236593<39>
233 71 4 2^233-1<71> = 1399×135607×622577×1168681298...31<57>
263 80 4 2^263-1<80> = 23671×13572264529177<14>×120226360536848498024035943<27>×383725126655170964501315730676446647<36>
313 95 4 2^313-1<95> = 10960009×14787970697180273<17>×3857194764289141165278097<25>×26693012026551688286164949958620483258358551879<47>
317 96 4 2^317-1<96> = 9511×587492521482839879<18>×4868122671322098041565641<25>×9815639231755686605031317440031161584572466128599<49>
401 121 4 2^401-1<121> = 856971565399<12>×2136958965524920285681<22>×594538100848945223169882301931953<33>×4743358775...93<55>
433 131 4 2^433-1<131> = 22086765417396827057<20>×737748363812546584876297<24>×5028667832511715101284999<25>×2707040788...21<63>
439 133 4 2^439-1<133> = 104110607×127321491658223<15>×122551752733003055543<21>×8738801468...69<90>
461 139 4 2^461-1<139> = 2767×358228856441770927<18>×7099353734763245383<19>×8461346092...33<99>
479 145 4 2^479-1<145> = 33385343×6293443049<10>×6834814451...61<57>×1086918660...81<71>
503 152 4 2^503-1<152> = 3213684984979279<16>×12158987054135300783<20>×1873030665061080894263<22>×3578015615...77<96>
509 154 4 2^509-1<154> = 12619129×19089479845124902223<20>×647125715643884876759057<24>×1075116828...69<104>
547 165 4 2^547-1<165> = 5471×172720604638150729<18>×157552196128597543991<21>×3094370432...83<124>
571 172 4 2^571-1<172> = 5711×27409×6969336604...27<73>×7084851186...39<91>
593 179 4 2^593-1<179> = 104369×3061144307110551703729<22>×4027016481761119763553952463965455549881<40>×2519699187...11<113>
601 181 4 2^601-1<181> = 3607×64863527×6433907608...93<60>×5513226964...63<110>
617 186 4 2^617-1<186> = 59233×68954123297<11>×1577519781...01<51>×8441284558...71<120>
619 187 4 2^619-1<187> = 110183×710820995447<12>×1093786816...39<66>×2539567680...33<105>
659 199 4 2^659-1<199> = 1319×11527429277532648241<20>×626564962613678012662146877852049<33>×2510867382...97<144>
673 203 4 2^673-1<203> = 581163767×41283139633378645724930694480520226273492263<44>×6839676957...19<59>×2388252511...09<92>
719 217 4 2^719-1<217> = 1439×772207×7375728433...69<51>×3364863383...51<157>
733 221 4 2^733-1<221> = 694653525743<12>×14399141148866077141941966959399761<35>×13948432601001273785699253868326270649<38>×3238594546...33<138>
739 223 4 2^739-1<223> = 184603056517613273120809<24>×48050683584092004380805463790111<32>×2561503388...63<82>×1272721565...51<87>
757 228 4 2^757-1<228> = 9815263×561595591×5722137022...93<79>×2403382164...59<134>
787 237 4 2^787-1<237> = 9951597611230279<16>×96512008100928793<17>×1711247935...03<63>×4952443062...47<142>
811 245 4 2^811-1<245> = 326023×1205618453...21<66>×2311981238...63<83>×1502740709...43<92>
823 248 4 2^823-1<248> = 1460915248436556406607<22>×1534086200463688788034864584049<31>×1431846377...47<95>×1743071575...67<103>
857 258 4 2^857-1<258> = 6857×2893063747...27<60>×6625204725...91<75>×7311640476...79<120>
859 259 4 2^859-1<259> = 7215601×13183242101527799620391251382520507315529510625017<50>×5860989871...37<62>×6894464928...03<141>
883 266 4 2^883-1<266> = 8831×63577×258777491057348926546569104663<30>×4438638313...47<228>
907 274 4 2^907-1<274> = 1170031×3256645177<10>×131895016205024767<18>×2152830659...63<241>
911 275 4 2^911-1<275> = 1823×26129303×201955048939840841121786425435009<33>×1799520676...07<232>
1013 305 4 2^1013-1<305> = 6079×71486759597273<14>×4112091256...73<59>×4912076350...01<229>
1031 311 4 2^1031-1<311> = 2063×435502649×8588431101...31<74>×2982096305...51<225>
1049 316 4 2^1049-1<316> = 33569×459463×1059099980653317121<19>×3692661882...53<288>
1051 317 4 2^1051-1<317> = 3575503×2157380128...49<63>×3050179060...99<69>×1025501468...99<180>
1069 322 4 2^1069-1<322> = 17481727674576239<17>×37473613084215372416028665821312617743022228898297<50>×5557036167...21<70>×1737449627...77<187>
1109 334 4 2^1109-1<334> = 30963501968569<14>×85608965982066833903<20>×2461601921...19<146>×1065805713...67<156>
1129 340 4 2^1129-1<340> = 33871×833798113×2682863551...11<139>×9624487049...87<188>
1153 348 4 2^1153-1<348> = 267497×84755607199<11>×1012236096...93<89>×5331087677...29<243>
1163 351 4 2^1163-1<351> = 848181715001<12>×337097300570078978047<21>×1042816042...87<73>×4201777709...63<246>
1223 369 4 2^1223-1<369> = 2447×31799×439191833149903<15>×4226520786...73<346>
1307 394 4 2^1307-1<394> = 101286184577<12>×141196558805510033914433414063<30>×161633497146742177992711798481<30>×1208644664...17<325>
1399 422 4 2^1399-1<422> = 28875361×4320651071020341609502042221583629017824960697<46>×9729831901...81<61>×1139677513...31<308>
1409 425 4 2^1409-1<425> = 264136919719<12>×34649857787874317478304238154673687249<38>×1484993916...43<64>×1042338191...67<313>
1471 443 4 2^1471-1<443> = 652662073639<12>×241722486510605423<18>×6149161897...13<68>×6734452045...27<346>
1543 465 4 2^1543-1<465> = 101839×472159×18333134391270351479674844445819664111063485906599<50>×3499807753...93<405>
1997 602 4 2^1997-1<602> = 395407×2982351597692070261023<22>×342849716969843931264330628335073<33>×3549722063...07<542>
2351 708 4 2^2351-1<708> = 4703×1357402684...91<52>×1705752023...79<55>×4836377904...41<599>
2381 717 4 2^2381-1<717> = 4902754929329<13>×6680044871611436077197194505521<31>×901939223649310421767556825677135846954351070231<48>×1914356587...69<626>
2549 768 4 2^2549-1<768> = 4363889×5490243876052618831<19>×324013067177330948216089046508086969<36>×2725405265...41<707>
2837 855 4 2^2837-1<855> = 22697×138871151×336396730297<12>×9923512934...69<830>
3037 915 4 2^3037-1<915> = 18223×145777×25596537003247670904155819094278523616227057000569<50>×2486604612...29<856>
3089 930 4 2^3089-1<930> = 279481049759<12>×91331189000126892893989706291353<32>×93262507747140213920830364839855613859777479<44>×3198733359...67<843>
3203 965 4 2^3203-1<965> = 24815594831<11>×592610284910292013073<21>×2897244767863537806877825949779988909058991<43>×3711906689...79<891>
3307 996 4 2^3307-1<996> = 10609622676254977<17>×2034656323703532334145890988609<31>×3413215452...03<55>×4353524133...13<895>
3413 1028 4 2^3413-1<1028> = 63148246745489<14>×708502447412410598209<21>×1801386698628063669444450416074615884859831<43>×3228981802...61<951>
3467 1044 4 2^3467-1<1044> = 4913650657744912439<19>×44206619588662227998657<23>×1036855844...07<55>×2081541164...07<949>
3637 1095 4 2^3637-1<1095> = 17075000822300799152287<23>×2914077909273961700066540881<28>×2720370087378938115190424396876883511195745489<46>×5183276147...37<1000>
3917 1180 4 2^3917-1<1180> = 407369×4606265160215972929759<22>×7679006328...37<61>×9459127487...73<1091>
5087 1532 4 2^5087-1<1532> = 40697×1678711×114097014833<12>×2803982690...57<1510>
5227 1574 4 2^5227-1<1574> = 1129033×47126633×227482218017<12>×2516905056...79<1549>
5393 1624 4 2^5393-1<1624> = 32359×4525200809...17<54>×1270133764...53<58>×1532089747...49<1509>
6199 1867 4 2^6199-1<1867> = 61991×743881×3895469424045161025776010136475884556282201<43>×6769416143...97<1813>
6337 1908 4 2^6337-1<1908> = 867442823536500063364056974417<30>×17816315586138579128417854744231<32>×291260082159225422294046286277013181681110191<45>×9413319504...03<1802>
7417 2233 4 2^7417-1<2233> = 118673×16269026327<11>×3888241452787718190543521<25>×7311578220...81<2193>
7669 2309 4 2^7669-1<2309> = 131093887×40807239817<11>×90406606411269877183013315812904972636799709663<47>×8213285483...43<2243>
8243 2482 4 2^8243-1<2482> = 16487×4207243687<10>×228591793996490954574648885834108681199807<42>×1549010340...29<2427>
9733 2930 4 2^9733-1<2930> = 2932747561<10>×353435802999708808999<21>×4424579967215442704801447<25>×1834385146...27<2876>
113 35 5 10384593717069655257060992658440191<35> = 3391×23279×65993×1868569×1066818132868207<16>
151 46 5 2854495385411919762116571938898990272765493247<46> = 18121×55871×165799×2332951×7289088383388253664437433<25>
163 50 5 11692013098647223345629478661730264157247460343807<50> = 150287×704161×110211473×27669118297<11>×36230454570129675721<20>
191 58 5 2^191-1<58> = 383×7068569257<10>×39940132241<11>×332584516519201<15>×87274497124602996457<20>
251 76 5 2^251-1<76> = 503×54217×178230287214063289511<21>×61676882198695257501367<23>×12070396178249893039969681<26>
307 93 5 2^307-1<93> = 14608903×85798519×23487583303<11>×78952752017<11>×1121774764...61<57>
337 102 5 2^337-1<102> = 18199×2806537×95763203297<11>×726584894969<12>×7877804732...69<68>
367 111 5 2^367-1<111> = 12479×51791041×78138581882953<14>×301311116540899114446723859201<30>×1975574008...81<56>
383 116 5 2^383-1<116> = 1440847×7435494593<10>×503823044204581129045587727<27>×15174923558680812616818436353130417<35>×240522700235167893496900256599634325263<39>
419 127 5 2^419-1<127> = 839×903780021613921<15>×5800422716722833271214743<25>×10287968884341772230096159036619433593<38>×29919490848598531825060153417921002916701815927<47>
449 136 5 2^449-1<136> = 1256303×6871197486841<13>×3578620616468306981503<22>×526385733768051189441947168504909159<36>×8939669339...41<59>
541 163 5 2^541-1<163> = 4312790327<10>×6115209994009<13>×77146448294831869472151022106713<32>×904106555381898703253733254107256591<36>×3913112911...79<73>
557 168 5 2^557-1<168> = 3343×21993703×4565508951414453792719<22>×22053236096920219270632521586535711<35>×6372521752...11<101>
569 172 5 2^569-1<172> = 15854617×55470673×182602768015690099110572536951<30>×110582329556343704552404016904356632250881<42>×1088066029...41<87>
661 199 5 2^661-1<199> = 1330270433<10>×3370159489168519<16>×1009157848082361225065617<25>×118420287267066844820208926433723871<36>×1785876939...59<116>
677 204 5 2^677-1<204> = 1943118631<10>×531132717139346021081<21>×978146583988637765536217<24>×5362511269...67<53>×1158330612...99<98>
691 209 5 2^691-1<209> = 906642603313<12>×10488160032325844521<20>×2833637724427940664433391497<28>×16636201944470267267182958207<29>×2291886288...41<122>
761 230 5 2^761-1<230> = 4567×6089×738686421813192728921171408273447<33>×2802302669...07<69>×2107048624...13<121>
821 248 5 2^821-1<248> = 419273207×109840721427977<15>×286121480219517473<18>×2534833058...23<62>×4186644906...71<146>
839 253 5 2^839-1<253> = 26849×138561000316919<15>×377801626929390823<18>×113910681722635191781067775764311<33>×2289649272...09<185>
853 257 5 2^853-1<257> = 2065711807<10>×513740645819473<15>×4727997228805279065233<22>×1844945010...89<80>×6488017349...13<132>
863 260 5 2^863-1<260> = 8258911×169382737×175642891399<12>×2928599996...01<97>×8546802933...99<137>
887 268 5 2^887-1<268> = 16173559×139714099995626040378049<24>×2382154297313396649423300881<28>×1958905756...79<73>×9785361819...03<137>
929 280 5 2^929-1<280> = 13007×340388595097<12>×9123887298...77<51>×1213545181...77<61>×9257164238...21<153>
937 283 5 2^937-1<283> = 28111×2419437071<10>×3481316282...43<57>×4665472298...69<59>×1051662080...73<154>
967 292 5 2^967-1<292> = 23209×549257×1438495818172960049<19>×73214830052905408299354903681341531907328727<44>×9291078206...73<219>
991 299 5 2^991-1<299> = 8218291649<10>×41473350001<11>×231620367206687<15>×6721885469...59<127>×3943730076...91<137>
1019 307 5 2^1019-1<307> = 2039×75407×8243323067...89<53>×1140356877...41<76>×3886812454...31<171>
1021 308 5 2^1021-1<308> = 40841×795808241×51731968308257<14>×1866013803867738316856548085894075729799351216503<49>×7162207820...01<232>
1033 311 5 2^1033-1<311> = 196271×36913223×88910338973359434076776038119074340716281<41>×163771629900828769671363963472861890953631838273<48>×8724820123...79<210>
1093 330 5 2^1093-1<330> = 43721×111487×42565880168703357316051447<26>×4611633294...49<122>×1109058770...11<173>
1181 356 5 2^1181-1<356> = 4742897×196834168670106625096599487<27>×199668911054466914791190422986407<33>×1808422353...87<73>×9742499217...01<218>
1321 398 5 2^1321-1<398> = 7927×1394977×4848071×15514776584693313272539134504494867053743089873<47>×5503435352...43<335>
1693 510 5 2^1693-1<510> = 10159×3361912721093432406946778492953<31>×414073707708708359213580367115039<33>×21637859486326546740118800926236362073<38>×1438975212...39<406>
2029 611 5 2^2029-1<611> = 2174887297707195387137<22>×16862485737890294913967<23>×1673313842828044768165904644769<31>×8920333613...77<58>×1126015449...93<480>
2069 623 5 2^2069-1<623> = 326903×1816583×1662927218441<13>×1776510807594595616100267889706057<34>×3863187447...47<566>
2251 678 5 2^2251-1<678> = 400679×778847×17620954939878356226435007<26>×1687942505818611032423917201<28>×4475941653...17<614>
2441 735 5 2^2441-1<735> = 38675902127<11>×152924392897<12>×343446375010967<15>×5349445835238920715559398496149751109514197898977<49>×5999768130...31<650>
2909 876 5 2^2909-1<876> = 110543×46174737359<11>×29101424767901148287<20>×2810063687215981074703243943<28>×1190395540...83<814>
4219 1271 5 2^4219-1<1271> = 1634011915849<13>×374348373815829857290734641903689<33>×258085857757974624851934511097909374553<39>×114993958477860428006858637956899699352321<42>×6117603206...59<1146>
4349 1310 5 2^4349-1<1310> = 12100966474679<14>×183495251103546507760826263409<30>×35797232484163710163467490373510908687<38>×47571370031120172757491816499652518228991<41>×3997708692...53<1189>
7039 2119 5 2^7039-1<2119> = 1252943×1057032553<10>×8541573097<10>×218216841131937276721<21>×3611550365...69<2074>

yangchuanju · 发表于 2022-10-17 05:03

最小的已被完全分解的280个梅森数（按因子个数排序）：
p 位数因子数分解式
223 68 6 2^223-1<68> = 18287×196687×1466449×2916841×1469495262398780123809<22>×596242599987116128415063<24>
239 72 6 2^239-1<72> = 479×1913×5737×176383×134000609×7110008717824458123105014279253754096863768062879<49>
359 109 6 2^359-1<109> = 719×855857×778165529×65877330027880703<17>×370906580744492785430299503112990447<36>×100361196281293745682520861860411315001<39>
463 140 6 2^463-1<140> = 11113×3407681×448747600991881<15>×239932071009857681156251129<27>×385606580062688087218266143<27>×1514826404...17<62>
587 177 6 2^587-1<177> = 554129×2926783×39483330766889<14>×73208283304744901303<20>×3728298863422039632638351<25>×2898153126...33<107>
641 193 6 2^641-1<193> = 35897×49999×1173835097<10>×2401258891949526685926151441<28>×7452763007...99<69>×2420161564...79<79>
653 197 6 2^653-1<197> = 78557207×289837969×16008527538753578495897<23>×3044938125828889184039273431<28>×3786707123184812723340717607<28>×8893122618...73<103>
773 233 6 2^773-1<233> = 6864241×9461521×1165626988664823792549971253972697<34>×101625278121114813541046756104385520302531369<45>×3102804258...07<52>×2081217395...81<91>
827 249 6 2^827-1<249> = 66161×1637241673<10>×1656489332706171370072190422631<31>×2550707823005230793212015514781257<34>×1339459522...99<80>×1459864713...23<93>
877 265 6 2^877-1<265> = 35081×1436527×1839699736969340949026772071177<31>×4478527345975241075738524755872079528958635991<46>×9138345796...23<71>×2655649011...53<107>
947 286 6 2^947-1<286> = 295130657×25749931927<11>×4621646208862937<16>×11892980076500863942962129100776937<35>×4247385134...73<104>×6705118430...89<113>
953 287 6 2^953-1<287> = 343081×562070136841<12>×40496650491842199555367<23>×824501439026637805612489876810823<33>×2106041424...21<60>×5614631499...11<155>
977 295 6 2^977-1<295> = 867577×1813313×2069655374719577273<19>×49858990580788843054012690078841<32>×2383040575...51<62>×3301809434...97<171>
1091 329 6 2^1091-1<329> = 87281×551978359×791343585595817<15>×97535130364653673535422113471207017<35>×496474000537225566899310934393330825097522839<45>×1437013471...83<221>
1151 347 6 2^1151-1<347> = 284278475807<12>×81344898763887260484533897<26>×658622107254745112979229913<27>×39146265717288735265327499826695585290297<41>×8311919431...47<103>×6171954834...79<139>
1187 358 6 2^1187-1<358> = 256393×113603023×1761533117965696212626325049982897<34>×766302834531296176581826196660125454171041129<45>×1533278332...69<63>×3486555298...69<204>
1201 362 6 2^1201-1<362> = 57649×1967239×8510287×2830858618432184648159211485423<31>×9303484663...59<61>×1354777128...99<253>
1289 389 6 2^1289-1<389> = 15856636079<11>×108817410937<12>×827446666316953<15>×9580889333063599<16>×16055826953448199975207<23>×4852621859...33<314>
1373 414 6 2^1373-1<414> = 1957392737<10>×4540035943<10>×95805718487<11>×416034935039<12>×1422039594...07<51>×4092737223...51<322>
1783 537 6 2^1783-1<537> = 107022793×59767828889<11>×21991704205191358757046463<26>×60701317462845977755176049<26>×1943163990190458280022304032374692151<37>×3285277443...43<431>
2243 676 6 2^2243-1<676> = 22373457765765203287<20>×1621855608482099266985412989105443846107961<43>×33584291626555988192435097466513894803912161<44>×25505344904088339250306780569062170259580500303<47>×6669478815648360567032338704138470582022170609473<49>×7828448307...39<475>
2447 737 6 2^2447-1<737> = 8163193×2771915076094391<16>×459390495235738452861593<24>×24700068031700569654277801<26>×3922634957266302331319319913<28>×4142870475...81<638>
2699 813 6 2^2699-1<813> = 5399×307687×1187561×7570504839257<13>×1987104667810711<16>×1017499748...17<770>
2749 828 6 2^2749-1<828> = 45737863×47065475057<11>×475325609449<12>×71379967497493020878447720712937<32>×2012663373...69<65>×2312815884...93<702>
2789 840 6 2^2789-1<840> = 128664552119<12>×36893679273343<14>×646806236672569040403110057249<30>×1175852797876979631858722975195410428697<40>×21150325778215526508794829601175050165456177<44>×4895564866...43<703>
4021 1211 6 2^4021-1<1211> = 10800407×2192551772209<13>×2889358374295985266660480623679<31>×1805978952876778448258839710221983<34>×8755191763...49<66>×2555305483...89<1062>
4751 1431 6 2^4751-1<1431> = 268982617×3274778783<10>×629530076753<12>×81630665742097<14>×1507074535068001<16>×2288751610...97<1372>
5233 1576 6 2^5233-1<1576> = 994271×9223417954129<13>×32101013028243569<17>×93603692660420120110355562102857<32>×2913486798065813495660442702490836503<37>×2428543426...51<1472>
7013 2112 6 2^7013-1<2112> = 54547899457<11>×1446087342943764981983<22>×1462102413595350700488992078837779280887<40>×1932091091898893820676219206120678104833<40>×1138366928030375456183260866527655539096399389159<49>×5237202577...49<1953>
491 148 7 2^491-1<148> = 983×7707719×110097436327057<15>×6976447052525718623<19>×19970905118623195851890562673<29>×3717542676439779473786876643915388439<37>×14797326616665978116353515926860025681383<41>
743 224 7 2^743-1<224> = 1487×1219280833<10>×14904366017<11>×118722715461092305629361<24>×4721525455401597740684262559<28>×4977047949106985392753791512048265888682003683833<49>×6137204207...03<102>
797 240 7 2^797-1<240> = 2006858753<10>×54573369937<11>×104757762864135516671<21>×747609928190821086986322983<27>×21023711323746974956423747989180911<35>×2896793131...57<55>×1595588224...01<85>
1097 331 7 2^1097-1<331> = 980719×4666639×22926719175799<14>×1734492765338661710171833<25>×36079584237927411511333522577<29>×205295973898620029528272537868263<33>×1259505554...43<220>
1361 410 7 2^1361-1<410> = 8167×3397057×137450113×30322542339673<14>×15459763525875943<17>×856450061281312036458486309832227737<36>×3287328502...31<326>
1723 519 7 2^1723-1<519> = 17231×56421359×381913289×5189864109981751703<19>×782974566556895066995270439431<30>×1689424363...69<51>×1854903578...91<400>
1999 602 7 2^1999-1<602> = 1807097×631388687840018727119<21>×63252453025264598101995353<26>×5347162249018972109619200063610257<34>×1835925785367240799404436249922514003983<40>×7452018296729329082588085050101877014364039300287<49>×1087309798...49<428>
2087 629 7 2^2087-1<629> = 561653441×7784142689<10>×13972903680989405047308631<26>×46569645515600223410122662599<29>×700690973459435662005774886079<30>×1144261721186669289407300288576366847993924905671<49>×7788973897...63<478>
2677 806 7 2^2677-1<806> = 364073×465799×502012457×154307510402911<15>×585294986690122657135969<24>×81699346406928829474760749455463<32>×1146083931...17<717>
9901 2981 7 2^9901-1<2981> = 87770464009<11>×4512717821471308759<19>×8336998551279784091551<22>×1017688752041649660766793<25>×25146117302614435382787771401<29>×1502440689076527620360606617623599<34>×2479176932...53<2844>
431 130 8 2^431-1<130> = 863×3449×36238481×76859369×558062249×4642152737<10>×142850312799017452169<21>×1807482391...57<70>
701 212 8 2^701-1<212> = 796337×2983457×28812503×1073825104511<13>×9983923992673<13>×15865578195367<14>×40686928318417<14>×2220668173...89<140>
1009 304 8 2^1009-1<304> = 3454817×198582684439<12>×20649907789079<14>×21624641697047<14>×30850253615723594284324529<26>×1134327302421596486779379019599<31>×3280162939...47<89>×1560053963...37<116>
1087 328 8 2^1087-1<328> = 10722169×2144921195591<13>×33990250708335239<17>×2105485879475562449<19>×421733524843190780249388261839156579767<39>×23864222009193938317456687011256092421563991<44>×2664797814...17<61>×3756253784...67<131>
1489 449 8 2^1489-1<449> = 71473×27201739919<11>×51028917464688167<17>×13822844053570368983<20>×1221632661...71<51>×9590951829...27<53>×4307681208...91<60>×2474373089...59<235>
3733 1124 8 2^3733-1<1124> = 1314017×2563787071<10>×29802151657<11>×336775122002089<15>×2369508169644641<16>×1679771749768055536667924144383889<34>×1960776513...17<53>×2106525274...57<983>
397 120 9 2^397-1<120> = 2383×6353×50023×53993×202471×5877983×814132872808522587940886856743<30>×1234904213576000272542841146073<31>×6597485910270326519900042655193<31>

yangchuanju · 发表于 2022-10-17 05:04

本帖最后由 yangchuanju 于 2022-10-17 09:48 编辑

如何寻找梅森素因子
基本方法是试除法，已经知道梅森素因子都是2kp+1型的整数，试除过程中只取2kp+1型奇数即可，直至试除到2^p-1平方根以内的那个最大的2kp+1；
如果找到了那个梅森数的一个素因子q1后，即可将该素因子从梅森数中除去q1，向下继续试除时试除到(2^p-1)/q1的平方根以内的最大的2kp+1；
在找到那个梅森数的第二个素因子q2后，再次将q2除去，向下继续试除时试除到(2^p-1)/q1/q2的平方根以内的最大的2kp+1；
直到分解到最后的余因子是素数为止。

该试除法是一种最笨的方法，试除繁杂且漫长，甚至无法试除下去，如梅森数2^1213-1是一个366位整数，
已经知道它有一个最小素因子327511，若用试除法，需试除135次方可得到；但要再用试除法寻找第二个63位的素因子，需再试除2.94724*10^59次，你做得到吗？
该梅森数尚有一个至今未被分解的297位的复合因子，再用试除法还需要大约10^145次，你及你的子子孙孙试除到地球毁灭能够把它分解开吗？
2^1213-1<366>=327511×7150798418...71<63>×6022881435...11<297>

ysr · 发表于 2022-10-17 08:22

yangchuanju 发表于 2022-10-16 20:47
经统计，在前1229个指数n小于10000的梅森数中，共有22个梅森素数，其余都是合数；
在那些合数之中，有280 ...

1277和1279都是素数，2^1277-1=2601983048666099770481310081841021384653815561816676201329778087600902014918340074503059860433081046210605403488570251947845891562080866227034976651419330190731032377347305086443295837415395887618239855136922452802923419286887119716740625346109565072933087221327790207134604146257063901166556207972729700461767055550785130256674608872183239507219512717434046725178680177638925792182271
才385位数，不知道为啥不能分解，有空了研究一下吧。

ysr · 发表于 2022-10-17 08:27

2^1279-1=10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087
这个才386位，网上软件也不能分解，有空研究一下试试。

yangchuanju · 发表于 2022-10-17 08:51

本帖最后由 yangchuanju 于 2022-10-17 09:39 编辑

ysr 发表于 2022-10-17 08:22
1277和1279都是素数，2^1277-1=260198304866609977048131008184102138465381556181667620132977808760090 ...

2^1213-1是一个尚未被完全分解的最小梅森数，
2^1213-1<366>=327511×7150798418...71<63>×6022881435...11<297>
63位素因子是
715079841864486572798550389025242773233223013545064445043124471
297位复合因子是：
"602288143593720536806089286738709175972359125227761761564073227910379498285174671775644305722270614135100412938071141915024322878163325882047318951741757889916312348536315125683454292668478299147363953500168798099737705487101263107775268036487596518883077658199869205832214060428355682611836617711"
王彦会老师如有兴趣和分解能力，不妨试一试这个297位复合因子。

请老师注意，297位复合因子的较小素因子一定要大于63位的第二个素因子；可用第二素因子以后2426k+1型奇数做试验。

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梅森数及梅森素因子

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