OddPerfect.org
Factors of Vanishing Fermat Quotients

Pace Nielsen has proven in a preprint that any odd perfect number must have at least nine distinct prime factors. In proving this, he found it helpful to prove that there are at least two factors greater than 1011 for some numbers of the form qp-1-1 where the number is divisible by p2. The number is always divisible by p by Fermat's Little Theorem, and the quotient after that division is known as Fermat's Quotient. Cases where the Quotient itself is also divisible by p are sometimes called Vanishing Fermat Quotients because the quotient mod p equals 0. The most famous Vanishing Fermat Quotients are for the base q=2; These are known as Wieferich Primes. However, only odd prime bases are of use to the odd perfect number proofs.

This list of vanishing fermat quotients with odd prime base and exponent up to 1011 was taken from Keller and Richstein's list plus an extension by Michael Mossinghoff. Pace is interested in factors where the exponent is the multiplicative order; these are often less than the full fermat quotient.

We are attempting to find two factors greater than 1011 whenever possible. This page lists all cases where two such factors are possible and lists the factors presently known. A separate page covers the cases where two such factors do not exist.

Factoring extent for the small composites is being tracked on another page. Please report any newly found factors on the thread at the Merserenne Forum.

## How to Read the Table

Consider the row that starts 59, 2777. These two entries mean that that 592776-1 is divisible by 27772.

The next column has the factorization of 1388. 1388 is the order of 2777 in 59. This means, among other things, that 591388-1 is the smallest power of 59 that is divisible by 2777. Pace wants two large factors of this number. The factorization of 1388 is provided to make it easy to find the algebraic factors.

The first row of the fourth column starts 347- and is followed by a large prime. This means the large prime is a primitive factor of 59347-1, which is itself an algebraic factor of 591388-1.

The second row of the fourth column starts 694+ and is followed by a large prime. This means the large prime is a primitive factor of 59694+1, which is itself an algebraic factor of 591388-1.

 Base Exponent Order Factors 3 1006003 2 * 3^2 * 55889 3*55889M 154680726732318637 9*55889L 103844037466916840539 5 20771 5 * 31 * 67 31- 625552508473588471 67- 604088623657497125653141 5 40487 2 * 31 * 653 31- 625552508473588471 653+ 211649260295455220087 5 53471161 2 * 3^2 * 5 * 148531 45+ 60081451169922001 445593+ 5810497963747 5 1645333507 2 * 3^3 * 30469139 30469139+ 52082118058261 30469139+ 481229581367 5 6692367337 2^3 * 3 * 278848639 278848639- 8930008316757509 278848639- 2323366860149 5 188748146801 5^2 * 239 * 1974353 239- 40093613041379 239- 1473534596915206322445556077174781171340308026061819537080103719298166168947549642538525464219037490151421698581086237756602026720817756235926209843391 7 491531 5 * 13 * 19 * 199 199- 86539116653269014086961051020627012232284504261471 199- 3742361194240057893199566966355314018920268076528360256893169784289227626965547117 13 863 2 * 431 431- 35910496578500372495225262919339090613 431+ P469 = (13^431+1)/(14 * 863^2 * 68099) 13 1747591 3 * 5 * 13 * 4481 39-    57745124662681 65- 158943831041162255277151 17 46021 2 * 5 * 13 * 59 59- 1365581260423071390161 59- 90008517325328860435221505121015340220148461 17 48947 24473 24473- 63895279579889 Sufficient because base is a Fermat Prime 17 478225523351 2 * 5^2 * 9564510467 25+ 4064228544226537005066401 47822552335+ 2008547198071 19 137 2^2 * 17 17- 99995282631947 17+ 274019342889240109297 19 63061489 2^4 * 3^2 * 7 * 73 * 857 73- 391818505243975817655620850033223 73- 3226690707486553756833988409595477959 23 2481757 2^2 * 206813 206813- 3783577742004112625957 413626+ 474060067609 23 13703077 2^2 * 3^2 * 380641 380641- 247968579451 3425769+ 3881067403177 23 15546404183 37 * 5507 * 38149 37- 1925658337781 37- 5713839242138307627889538424597962861 31 6451 3 * 5^2 * 43 25- 20235942281002951 43- 3049055684506560663410351046998584180840895763387409 31 2806861 2 * 5 * 7 * 41 * 163 41- 42481797154433176612759 41- 132259604354473376342663326676479453 37 77867 2 * 38933 38933+ 605933589769 38933+ 493744755578369298257 37 76407520781 2^2 * 5 * 3301 * 1157339 3301+ 3643316113499743 3301+ 373684972381498348496022049 41 1025273 2^3 * 128159 41 138200401 5^2 * 13 * 2953 65- 1648439718668446778143137761131213836674769976364551997360615081514325961 191945- 803013424201 43 103 2 * 3 * 17 17- 3807926835707 17+ 27147048848953409 53 59 29 29- 39392783590192547 29- 88148880022265333 53 97 2^4 * 3 12+ 62259682520881 24+ 153154713757537 59 2777 2^2 * 347 347- 1577618183226146053681303 694+ 156037307738480227937 67 47 2 * 23 23- 159298895525201570753486381 23+ 6652974112233411152741142680306351347 67 268573 2 * 3 * 22381 67143+ 12095800707121 22381+ 12988499163955331 71 331 3 * 5 * 11 55- 143554218709131407 55- 1401479667198929984062185352247019146137575004341891557131 79 263 2 * 131 131- 761125627205909375604086348842011105513919 131+ 191019285505467760133999 79 3037 2^2 * 3 * 11 * 23 11- 1750258119644519 33- 11409584517192577 79 1012573 2^2 * 3^2 * 11 * 2557 11- 1750258119644519 33- 11409584517192577 79 60312841 2^3 * 3 * 7 * 3779 21- 387782571085603 21+ 686421384890977 83 4871 487 487- 288751256161595856579468839 487- 539654131782457562603348716181223730483409 83 13691 5 * 37^2 37-    77294079343261321938167 37- 810401974611817725183817038020767387 83 315746063 2 * 2153 * 73327 2153- 24292265339999 157873031+ 221336410731691 97 2914393 2^3 * 13 * 9341 13- 8224356155341457 13+ 237393489259057 97 76704103313 2^3 * 4794006457 4794006457+ 44469203895133 9588012914+ 3394156571557 101 1050139 2 * 3^3 * 19447 27- 1653418568375032120019428396177 27+ 39312028293482485758634561 103 24490789 2^2 * 3 * 7^2 * 41651 49- 330773078230085653621541237 49- 10462447282739821751009564489723431427910390166514177110901 107 97 2^5 * 3 12+ 17181861667239601 24+ 352484591145491846304337 107 613181 2^2 * 23 * 31 * 43 23- 154317473175739320798279261750709 31- 4756744811420477568753648619 109 20252173 2 * 3 * 37 * 45613 37- 9057766586713846275063638613129747866622164083957969 37+ 995309568661550063695233580057386307707058100503 127 907 2 * 3 * 151 151- 141051780820060939 151- P285 = phi / (121822452850129 * 141051780820060939) 127 13778951 5^2 * 275579 25- 373933551512831586851376055846423651 Need Another 131 754480919 2 * 19 * 19854761 19- 139941921745500859 19- 1627898501375482741 137 29 2^2 * 7 7+ 152649866251 14+ 1260297499721989 137 6733 2^2 * 3 * 11 * 17 11- 2346320474383711003267 17- 8859813646194068340402291825431 137 18951271 3 * 5 * 13 * 48593 39- 931659959992945570101408381932879311 65- 3625814360634980900585896585178375841931363771006550091150789272055831799954306545336207417040426261441 137 4483681903 3 * 19 * 61 * 30703 19- 291173513911804236660449587340421782827 61- 96488938834989542506408195841434628036925740132116494644176186842000542282442099707138348970440943271921297 149 29573 2^2 * 7393 7393+ 215871622567 7393+ 25276538747078803 149 121456243 3^2 * 113 * 211 * 283 211- 1838349798362881 211- P422 = phi / (10973 * 1936981 * 2788389899 * 1838349798362881) 149 2283131621 2^2 * 7 * 11 * 17 * 37 * 2357 7- 11016462577051 11- 81042426245204504653 151 2251 2 * 3^2 * 5^3 25- 7595719904010033008065603640626272322201501 25+ 2507620255217609317790427701 151 14107 2 * 3 * 2351 7053- 14270362576120775983 2351+ 3126740031140148447543103979 151 5288341 2 * 3 * 5 * 53 * 1663 53- 357397534980935741971325377623554898373083914885729043983632085869376882382403 53+ 737046472904406533195975350068315503524611818890351 151 15697215641 2^3 * 5 * 53 * 83 * 89209 53- 357397534980935741971325377623554898373083914885729043983632085869376882382403 83- 400507817476982154455389191772230422065057323604656051416898329466778018326508387326592108565286094593729821879486564745351170640336212056131091287298477487 157 122327 2 * 1973 157 4242923 2 * 2121461 2121461- 165363642029 Need Another 157 5857727461 2 * 3 * 5 * 13 * 43 * 174649 13- 281420912955937 43- 92117590758121432696113752546499673127113969776449130596399568476205337 163 3898031 2 * 5 * 251 * 1553 1553- 80820180096117884248711 1255- 60289798382803991 167 64661497 2^2 * 3 * 37 * 72817 37- 83248346139050180999166452317788265831752031043393683716160387495785560868179 37+ 464403289904565810584555667726453150470883620966341542135789665393289758383627 173 3079 3^3 * 19 19- 279227865151633389967 27- 1387915239820172935147327 173 56087 29 * 967 29- 9271231438769561 29- 244981203696414457 179 35059 3 * 5843 5843- 218990696626493221 5843- 9669301729139302268063 179 126443 191 * 331 191- 1684827183733210732987140071 191- P401 = phi / 1684827183733210732987140071 191 379133 2^2 * 13 * 23 * 317 23- 12070833277819 23- 709691990009063207 193 4877 2^2 * 23 * 53 23- 5900911239006733 23- 599580140353613008549474293409 197 653 163 163- 800339854680407 Need Another 197 6237773 2^2 * 1559443 199 77263 2 * 3 * 79 * 163 79-    22131645640847437 79+ 930771937874506753729546628340606952192699098893482989855225671644437122116037860294488895170954109830176491314036577905971379931963 199 1843757 2^2 * 14869 29738+ 761842169225273 29738+ 3234408137926822614557 211 279311 5 * 17 * 31 * 53 17- 473657018821793557815477348357239 31- 1054121948159195394769061642028988717474032847273 223 349 2 * 29 29- 13253060227619075636110997630748067 29+ 109062276310290150424370591574354167 227 40277 2^2 * 10069 10069+ 155625706960503133 20138+ 656369036254548078386389 233 157 2 * 3 * 13 13- 11336512831824701 13+ 104230022966279507207 233 86735239 2 * 3 * 773 * 18701 14455873+ 219960563569 773+ 21067084222266547538929 239 74047 2 * 7 * 41 * 43 41- 2402104740201909791429 41- 19780204581562816766126848164530247019813978275520029172841389 239 212855197 2 * 3 * 17737933 17737933- 162754404241501 17737933- 1125862083377 239 361552687 3 * 11 * 23 * 29 * 43 * 191 11- 26550464126812634441687 23- 1461029169764229202893941669928319 239 12502228667 6251114333 6251114333- 2137881101887 Need Another 241 523 3^2 * 29 29- 363140811956644727165114269418832750816589134197458265596018127 87- 1579758532183833562846140817437974196440079771856557372850867 241 1163 2 * 83 83- 29268685579993 83- P175 = phi / (27936473 * 29268685579993) 241 35407 3^2 * 7 * 281 21-    1893096722707 63- 1833905601838199806062088896199 251 421 2^2 * 3 * 7 7- 251059142817757 14+ 2014041263472963229 251 395696461 5 * 443 * 14887 257 359 2 * 179 179- 11323442498975992826664177106393 Need Another 257 49559 71 * 349 71- 694970559443441 71- 1863575510379660502209623595521569998628940719133525999183574191710309610857531653852234932703945148110061050197757940647732781318535758187 257 648258371 53 * 59 * 20731 53- 311980599694983423669679719669301746086303369 53- 926765428265324364431661428927670012452314526997715463196318128041273 263 251 5^3 25- 31202512205401 25- 354745666933328230951 263 267541 5 * 7^2 * 13 13- 539093310059453790343 49- 70669457745222930641719300601558385226306900383306737391976967 263 159838801 2 * 5 * 11 * 12109 11-    3124914562747294549 55- 3128907019617746878889901067038944061366583951 269 83 2 * 41 41- 178834672604566680942820618431015116485181076223549 41+ 10608360396807762453261859839365050334689580980396193033150879617557211216991881681 269 8779 2 * 3 * 7 * 19 19- 704269952391999908595285326878148728951 19+ 11934573486665131960229161 269 65684482177 2^6 * 3 * 29 * 307 * 19213 29- 150532558833384151 29- 518535270321995270856805206247 271 168629 2 * 42157 271 16774141 2^2 * 263 * 1063 2126+ 318984212960489 279569+ 79368593511941 271 235558417 2^3 * 29 * 197 * 859 29-    258570308924423 29- 6095590859278129374026417 271 12145092821 2^2 * 5 * 7 * 4691 * 18493 14+ 1305174058553 14+ 14752065295493 277 1993 2^3 * 83 83- 1539760247573 83+ 936835066163219 281 3443059 191281 283 46301 2^2 * 5 * 463 10+ 213141596441 463+ 7434594377381 293 83 41 41- 91813998019990093 41- 1015032763766944663920338548803102446764467699898818317404240272759983388967 307 487 3^5 9- 279067340218231 27- 33978340801612914202909610154622185607 313 41 2^3 * 5 10+ 302030240221226801 20+ 674537617504921 313 149 2^2 * 37 37- 163072747762875039437002169213 37+ 1385184514183084956550117156705944856722683074940478313961608167847184991782491 313 1259389 2^2 * 3^4 * 13^2 * 23 13- 240162718594025891 23- 2995183555466410410776093988201797 317 107 2 * 53 53- 1519680132684510090962349984494217935467402110425038434172839642059334570457385048198391116706736056826957 53+ 1501663628442024552556551058828070500584636833499094625583429652539848018157028808137405681 317 349 2^2 * 3 * 29 29- 11580833156868557339472630053843264905262377794251 29+ 61367058904217642755152093256495199 317 2227301 2 * 5^2 * 22273 25- 250470675769051 25 - 17526467471026912410930751 331 211 3 * 5 * 7 15- 364079103131761 35- 79459996550395924293461 331 359 179 331 6134718817 2^4 * 3^2 * 109 * 65141 72+ 2647811042004144905541244542191372641 109- 304533672616199149 337 30137417 61 * 61757 61- 10489350794570806864459917679 61- 60211362861502116632254368057553312563964310132449434314701 349 197 2^2 * 7^2 7- 1812169199976451 49- 14584330816732341411119943716563571881924625275054293185359523761 349 433 2^4 * 3^2 9- 16279070095441 36+ 3606863244954261906093847195422432889 349 7499 2 * 23 * 163 23- 1282994429098372621 23- 70754543192579591224955446564323593 353 8123 31 * 131 31- 2538292850741 31- 773588222741928947195283661694700870147359 353 465989 2 * 97 * 1201 1201+ 1450492427081940778189 97- 441700543814981841392909694018356229678497867567874488883075248473 353 17283818861 2^2 * 7 * 11 * 11223259 7- 1940350890330343 11- 66789352236795654577 359 23 2 * 11 11- 59577705183437736791 11+ 242729825559563 359 307 2 * 3^2 * 17 9- 112671246731059 9+ 5618775840823 359 24350087 17 * 179 * 4001 17- 85001215236190499 17- 898043687440549395595979 367 2213 2^2 * 7 * 79 79- P189 = phi / 258044263637 14+ 205869281458174762686805425173 373 113 2^4 * 7 7- 93115265278967 14+ 270687298347677 383 28067251 3 * 5^3 * 37423 15- 1587430225392031 25- 122584173128620581918373625551 389 373 3 * 31 31- 80627591959475835271737989 31- 7533661454672065622445205385645751606720569 389 29569 2^5 * 3 * 7 * 11 11- 79545183674814239059370551 11+ 1774835351741 389 211850543 105925271 105925271- 2740498611313 Need Another 397 279421 5 * 4657 23285- 1425481462015626551 23285- 49717784046818889557641 397 13315373041 2^4 * 3 * 5 * 5581 * 9941 15- 132458919591571 15+ 2421068129616151 401 83 41 41- 591316533225138384114054798581 41- 10954140430475464661151733514193169121 401 347 2 * 173 173- 1042198130362099766300376928177986191 173+ 85660987758281979674688326402829982118897 401 115849 2^2 * 1609 3218+ 10825870040077 3218+ 2281662728455331265637 409 34583 17291 409 1894600969 2^2 * 3 * 1283 * 4733 12144878+ 48722578662841 6072439+ 262098612119 419 173 2 * 43 43- 29289324923257419024299094089980312550622558572613593313925709668504890590772819283 43+ 17423662114746663604509446956283032619 419 349 2 * 3 * 29 29- 283819148579748177857006411440157515250788163978187258993063998381627 29+ 13315550643509564823996989838474064543710199583779 419 983 2 * 491 491- 143255450017752419 491- 13989089350123516432359468721757950233339581 419 3257 11 * 37 11- 4086571551344147723089 37- 16357689705294748931298930654427896012173369667926524786581884858319209282713591813 419 22891217 2^3 * 439 * 3259 1430701- 110799151015961 439+ 2945590425912007 421 1483 3 * 13 * 19 13- 585956616593534561051 19- 2486573758578350618342222456569364532017 421 350677 3^3 * 17 * 191 17- 30872424600517269383579721139923943997 27- 57538226342986860541001735395779146692186636981 431 12755833 61 * 8713 61- 343950623131231 61- 122645581479467763314241119744080221133135504336631200075911072492718885718004303319971825256301209184620150978951 433 129497 2^3 * 16187 16187- 179908756984145726011 Need Another 433 244403 122201 439 79 3 * 13 13- 141631501901853944278871 39- 30107380174838505723337135448737753 439 170899693 2^2 * 3 * 14241641 42724923+ 702568633813 42724923+ 1211083658102611 443 3406223 17 * 100183 17- 140771626575571446645499 100183- 218836838575793917 449 1789 3 * 149 149- 115014172581882179 149- P362 = phi / (5307083 * 44076883 * 115014172581882179) 457 919 2 * 3^3 * 17 27- 640012258318122164546749 27+ 33722134399343821720050964426363021 457 1589513 2^2 * 198689 397378+ 157110093907873535761 Need Another 461 1697 2^5 * 53 53+ 145727078501489 53+ 933434675936082845341979656822008947960973070725115739649390663075767870601126842365744351298487121807423 461 5081 2^3 * 5 * 127 10+ 20196866080328956541 20+ 293558183950864144227961 463 1667 7 * 17 17- 69432867826513 17- 67335065332476908719 467 29 2^2 * 7 7+ 145785376296533 14+ 127939479530790544999878367537 467 743 2 * 7 * 53 53- 6138386415329516022429203705120198394019097485836770241649763421956888663 53+ 2023157092887142695087879473018632214713984483510612521753336944402512621391446165065004571317033 467 7393 2^4 * 3 * 7 * 11 11- 494424256962371823779424877 11+ 240503826471577407297329 479 500239 2 * 3^2 * 27791 9- 635710644585019 83373+ 57782792023039 487 1069 2^2 * 89 89- 35505687955655611 89- 323807759567497987 491 79 2 * 13 13- 11290902796266652546651 13+ 31393527122337443214834723301 491 661763933 165440983 499 81307 3^2 * 4517 13551- 262329164194439119 13551- 2418601886578934487583 499 24117560837 2 * 8093 * 745013 6029390209- 2331678136357694989 Need Another 503 229 2 * 3 * 19 19- 2582484561590956831 19+ 41362397701447377151521984337 503 659 2 * 47 47- 6666637688308623973238358974732216821385025051654641364034192031807276905323396246435431 47+ 122118855928422856314722353766911443566374672078126149177038267 503 6761 2^3 * 5 * 13^2 13- 1593316347137624638260967 13+ 1134257918255890168844065327 509 41 2^3 * 5 10+ 164667774614948401 20+ 1973379117573727834750361 509 7215975149 11971 * 150697 521 8938997 2 * 11^3 * 23 * 73 73- 28389176436263 73- 127046866408637191 523 9907 2 * 13 * 127 13- 7917272313053609286926516514677 13+ 48084073702461781427 523 19289 2 * 2411 2411- 592096880773 2411+ 180544811908089283 547 1691778551 5^2 * 7 * 53 * 8291 53- 157807469272376914806938750637329512096719982148985701 53- 778188294192437874964897786162885332797948860494283606773667638266020317670780523 557 39829 2^2 * 3 * 3319 6638+ 140744518301 19914+ 9916685341669 563 18920521 2^2 * 3^3 * 5 * 17519 27- 199200848830036507 27- 15474726381276170437764673027 569 263 131 131- 74226203895619 131- 2188413654974639 569 25359067 3^2 * 677 * 2081 1408837- 516572627443 1408837- 4217297206021 571 23 2 * 11 11- 3867675755483 11+ 419116488063307 571 308383 2 * 103 * 499 499- 114002784427466383 499+ 1326542568304073849051 577 71 5 * 7 7- 1834838406941 35- 11895940705720965457099922478120080186357726271635862963031 577 1381277 2^2 * 13 * 101 * 263 13- 15448574625823 13+ 970816703270714843401 587 22091 2 * 5 * 47^2 47- 22808311866941385522390523090503262333088679096032607762406394343883578709101725775807420081917817322801806400443032864210285437 47+ 282257008997374983256877201759 587 6343317671 634331767 634331767- 567696479734193399 Need Another 599 35771 5 * 7^2 * 73 7- 46268622795238201 73- 5167531064864081 607 40303229 2^2 * 1439401 607 22035814429 2^2 * 3 * 149 * 701 * 17581 149- 5101963733414333867 447- 3044587194400738639 613 4073 2^2 * 509 509- 52563562568040983794422928907474899 Need Another 613 81371669 2906131 2906131- 2364666484343 Need Another 613 18419352383 9209676191 9209676191- 184193523821 Need Another 617 101 2^2 * 5^2 25- 1145624553182829328237193409226107251 25+ 195721941292532603214229975342951 617 1087 2 * 3 * 181 181- 9702726780407938649 181- 1016316514692800907301 617 6007 2 * 3 * 7 * 11 * 13 11- 10866472977391930428570371 13- 38592181531151807801943457426099 619 73 2^2 * 3^2 9- 3667303462519 9+ 123895768231 619 11682481 2^2 * 5 * 48677 10+ 21553810423954789956721 243385- 182384930681 619 52649183399 2 * 7 * 3760655957 26324591699+ 1948019785727 26324591699- 18637810922893 631 1787 2 * 19 * 47 47- 2149512027144932910975058305116283353 47- 304185208237244358320750017379571162495100965987217 631 5741 2 * 5 * 7 * 41 41- 1935813299146410294768533 41- 3056288003905025559691482914639205430291905753536068106320765180568315502856233719 641 24481 2^4 * 3^2 * 5 * 17 17- 11423150663625166960462613 17+ 21091704665701686749 643 307 2 * 3^2 * 17 9- 1239912343561501 17- 61525371110323961 643 859 2 * 11 * 13 13- 5002736399322454070215301405332201 13+ 80357055747826455072706212827 643 460609 2^5 * 2399 16+ 105579695041 16+ 4043571881790322556171751491105761 643 7354807 2 * 3 * 29 * 43 * 983 43- 2311800997464014898374467462271 43- 26242785364525583863151060937212373356000877327336767892621240608587 647 15266862761 5 * 43 * 8876083 43- 6058585050599360882855639967823 43- 63747398386573850666250762281255558690591955669147146398015556296782595963 653 1381 2^2 * 3 * 5 * 23 23- 16344257268406282067 23- 110482254005123138144435961784174522717087 653 22171 3 * 5 * 739 15- 13123058537131 3695- 10826960096231 653 637699 23 * 4621 23- 16344257268406282067 23- 110482254005123138144435961784174522717087 659 23 2 * 11 11- 77742873805239893361253099 11+ 807556971667909 659 131 5 * 13 13- 333609940477399 13- 6852450820603181 659 2221 5 * 37 37- 2636631908459346147831033803558174429038512917832803300721203229021264612138401497486361399 185- 6680777196371561 659 9161 2^3 * 5 * 229 10+ 35569873176487032158161 20+ 42395309058702526712206708020916841 659 65983 3 * 7 * 1571 10997- 14900291059669 32991- 32669613950653 661 441583073 2^5 * 7 * 23 * 85711 23- 5437415646713 23- 2785449070121546876012939697932929639031077 673 61 2^2 * 3 * 5 15- 121426283431 15+ 7768642762831 677 211 3 * 5 * 7 15- 30485129299404931 21- 16095633374482971163 683 1279 3^2 * 71 71- 10468866525017 71- 5341115714439713055494023775160841644601798991430593982089241222505769606479632088341637742412389747089646483014028398484538040370635957472515957703018819633041 691 509 2^2 * 127 127+ 6913021836871 127+ 1597368711978311 691 1091 2 * 5 * 109 109- 45275046236813 109- P286 = phi / (37233311 * 45275046236813) 691 9157 2 * 109 109- 45275046236813 109- P286 = phi / (37233311 * 45275046236813) 691 84131 2 * 5 * 47 * 179 47- 810939432247741328260303007456596109 47- 29839492745714861927402849640041798711539499056776665947710253 691 10843045487 2 * 7 * 13 * 101 * 589873 13- 43566718584923 13- 272402240126987379371 709 199 3^2 * 11 11- 32142180034067960734115528951 33- 246846925104177234876957391 709 1663 277 277- 88867987318576068105006245393497 Need Another 719 41 2^3 * 5 10+ 71421716410853446768321 20+ 513438124105345418842561 719 4414200313 2^2 * 3 * 183925013 367850026+ 201560478946493 Need Another 739 9719 43 * 113 43- 109315109978538599249 43- 9504688302309206841549948554866633850212075817292336395598470970459200287244293873363366571951 739 5681059 3 * 107 * 8849 321- 18944495006328739 321- 47649267831596519617 751 409 2^3 * 3 * 17 17- 251206628001493 17- 396231817958422167475388316923 757 71 2 * 5 * 7 35- 1252353759968265476969697055789781154157429634613247749718320411008401 35+ 564367225817823714704381799890831561 757 242789 7 * 23 * 29 29- 12755397830861341 29- 19269675845415341105480474280180113075891767 761 907 2 * 151 151- 40602891408274834862207861 151+ 594196348066391156135358197698769 769 1305827821 2 * 3^2 * 5 * 11 * 19 * 103 * 337 11- 11357164710613604107501 19- 12488732622361173231907884961451 773 787711 3 * 5 * 7 * 11^2 * 31 31- 2529726899183489 31- 2465784520316637015107 773 26259199 3 * 7^2 * 89317 7- 409084572289 49- 2495290020456418542852776464463910156290453201 773 142719149 11 * 13^2 * 17 * 1129 11- 319831381244161569949 13- 671577079433852248719505906321 787 427541 2^2 * 5 * 21377 10+ 1195508958401 106885+ 7581563399681 797 8273 2^2 * 11 * 47 11- 15747384825943 11+ 661764792107 797 14607661 2^2 * 243461 243461- 31970645070677 243461+ 84177692501521 809 59 2 * 29 29- 1820921058976765357538241234627268250723879603 29+ 302542547387622551039725602622193232017457369297318472980002190535339 809 448110371 5 * 2213 * 20249 20249- 240049204474873 2213- 15366043798475037570996955301308103 811 211 3 * 7 7- 14278950628867 21- 63728737863569353 821 83 2 * 41 41- 3547083078257280638236604630646226980017679569366201285233 41+ 27921262359223400561321835211035376131624516149884838495082503304894981770717051574019447477 821 233 2^3 * 29 29- 24643599200095736992767360124803668302697934493047340719151271670860182248643 29+ 128609109438087874878073146177745007 821 293 73 73- 110553401640370745848646039158681 73- 51630553390967519384493333888867540991079334850902138859281694249292745330994222988349215396854060391140951449458276317072905305908479002124692369 821 1229 2^2 * 307 307- 3866637344827 307- 13504150974307063663 821 37871 2 * 5 * 7 * 541 7- 63101741964353 541- 4269171570311939 821 209140301 2 * 443 * 4721 443+ 1098064261664720951 2091403- 4186707324070231 823 2309 577 577- 1684739430923662296081215062090807115949877Need Another 827 29 2^2 * 7 7- 405754222441 7+ 319527491618412043 827 9323 59 * 79 59- 101385095988644233 59- 54745095923089396230998368589 839 5227 2 * 13 * 67 13+ 121513968522143373592008111476296093 67- 37820627020672205459425871626061899363 839 11840951 2 * 5^2 * 11 * 21529 11- 3600456782848213993 11+ 14575508737099510332529 853 1125407 562703 857 157 2^2 * 3 * 13 13- 12276693188692165256869 13+ 95783106845694873574087 857 1697 2^5 * 53 4+ 269707666801 8+ 8557908885996415283153 857 32478247 3^3 * 200483 27- 130170009048992011230247 27- 631027055370227178175789033 859 71 2 * 5 * 7 35- 640635770552290316444968157394717290638852385651 35+ 2012742237153154587564784146436930621661608358550922022869531 863 467 2 * 233 233- 143314635814757558574889 233- 517848744527043681714953687877126923 863 12049 2^2 * 251 251+ 20244399735177072853 502+ 187209992537977 877 78926821 2^2 * 5 * 7 * 187921 10+ 11359912053172061581 14+ 736695871140428549374299601650017 881 23 2 * 11 11- 25066551862378295281 11+ 83166204178397077 881 22385723 11192861 11192861- 27337802079423499 Need Another 881 94626144313 2^3 * 7 * 563250859 7- 10886298318204709 7+ 2666826651077 887 607 2 * 3 * 101 101+ 186415992008777 101+ P268 = (887^101+1) / (888 * 209071 * 70300849 * 186415992008777) 887 60623 17 * 1783 17- 8390312489467 17- 71258053180146127727 907 3497891 2 * 5 * 11 * 31799 11-    5348869392189741991 11+ 244018275212131300278113 911 318917 2^2 * 6133 6133- 3322035750367 6133- 29747827427179 929 62199604679 2 * 31099802339 937 41 2^3 * 5 10+ 118835508290854605674861 20+ 121265491662021680201 937 113 2^4 * 7 7- 29778361446737 7+ 3330260911451531 937 853 2^2 * 3 * 71 71- 23378779190204219292649 71- 129150889328553619302387671 937 22343 2 * 11171 11171- 60540655285727807 11171- 112960633386972919 937 500861 2^2 * 79 * 317 317+ 27431675541501113 317- 55654879628760813917 937 1031299 29 * 5927 29- 33252388767007 29- 4867987312332055501463099210058722568698731382429028439106534632116563 937 258469889 2^10 * 63103 8+ 2418873425489 16+ 219771653381636209478908853598433 941 1499 7 * 107 7- 695023534042345747 107- 837176475407319401 947 5021 2^2 * 251 251- 4930017072892452022901 251- 85834753896881008093430138660158993013406926131 953 513405611 5 * 17^2 * 59 * 3011 17- 194740230586482119543728597 59- P167 = phi / 5035651 967 4813 2^2 * 3 * 401 6+ 874390502833 401+ 239329884229001 967 44830663 3 * 829 * 9013 829- 2194386049517 2487- 65301264170257 971 401 2^3 * 5^2 20+ 75009505237181235328347521 25- 207359136394007927024401 971 9257 2^3 * 13 * 89 13- 303231244032490217649645940267 89- 3830187325673 971 401839 2 * 3 * 66973 200919- 373703312431 66973- 1813997995177 971 7672759 3 * 727 * 1759 727- 107832403857149897 1759- 10943651174229352003 977 239 2 * 7 * 17 17- 7695349282294021778006851712082373624997 17+ 46600479435539023782173411 977 401 2^3 * 5^2 25- 4307139312468717998474572720974228676994651 25+ 186613963038398795973792253788630400601 977 37589 2^2 * 9397 9397+ 101168459087 9397- 1877238840353 991 431 2 * 5 * 43 43- 6101095635657562813536904175957431070032331779195936987 43+ 610603335100407574198463143814702268131694189555595341253433 991 26437 2^2 * 3 * 2203 2203+ 3800202779831 2203+ 18327166282153 997 197 2^2 * 7^2 7+ 108139574401741 49- 419215290992711174148390961430952312396980787493364880169056121329154166428430676812473404351452816643660762755588011 997 1223 13 * 47 13- 138869522626352012008697 47- 77434055998495993938155561355803226810783967