OP Home Composites Cleared Against Pomerance Fermat Quotients ElevenSmooth |
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.
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 Need Another |
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- 1684739430923662296081215062090807115949877 Need 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 Need Another |
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 |