Factorization of 13 consecutive 500-digit numbers N+k, k = 0 to 12 Discovered by Joe Crump and John Michael Crump, December 16, 2009 ************************************************************ Formula for N: x = 2^92+227683166; N = (2*x^3 - 3*x^2 - 5*x + 2)^2 * (4*x^6 - 12*x^5 - 11*x^4 + 38*x^3 + 13*x^2 - 20*x - 6)^2 / 16 - 12 Decimal Expansion of N: 12815419025806882624989107869626377225388252892348109143014972905583523571403186585248358925874769762143571465837894872353231546616266578261353426940275947134455203755973216086142017823730363315581887837636864244178669765557164623788557589071633903483494919111014070818792637091615757088898914231197548672788150791324639157778351397381132289522595508576244221687211708616979359070975986888468154087947303237066107215682723664525548690542907933932647424100409739619639031424304876201990797628540181044 Decimal Log of N: 499.10773281088082166 Prime factors of N: 2 x p1 = 2 p1 = 3 p7 = 1022689 p11 = 11589211679 p482 = 90106077567905848794679141405971174164718572711221915573022734945703704384793963745077778215996651287409891497024984902839691258008059449222017294433985202212228467235436071625539841323527705294369131263739320324684806194259027421578381712760246056325231341935213631919369902819564331896551488411622767454212729222382054962910191052191775766588232628196563643531271514671264349760680919717638233875022115388579915168189302993875999683830070842213573577385318863307863040894799100177 Prime factors of N+1: p1 = 5 p4 = 5261 p23 = 14307105795982965290513 p473 = 34052007167080758027828178073108723499270455401664898452219102401501831329053880733206655089332513991526206157730967297962704134966895819205984771331639362728152741238217905097907922484108876769069265310757244509613357477563918257990355946140151897982958895502074665019919380359168945788387549721448948049982930348866028976152458917202180313593773929992739413546148044778853276672441168530389019734805119793603192891954302489235588030269305091771910604862830126770276914213 Prime factors of N+2: p1 = 2 p2 = 13 p3 = 227 p4 = 1889 p493 = 1149480604757436813371633259385058947222155708614634507886351694366332071388994173696075867533465678083801030546562162789227000835085519660485425254476364987979526168998639691468685712026839231318334260867942428303428359836493378417860307474136312504585207507967534564356398652098960728505497524611673809040528633583095909541601531327289821408270456325402809294999165711292146085998607831969114209335441937481610904315458799040185809777711078543746502930645553715776514141091585736429333752557 Prime factors of N+3: 5 x p1 = 3 p2 = 11 p2 = 37 p2 = 73 p2 = 89 p4 = 1697 p5 = 22111 p5 = 24379 p5 = 50993 p5 = 75403 p6 = 294431 p6 = 791899 p7 = 7143163 p8 = 37162403 p8 = 40238771 p9 = 318066373 p9 = 774022187 p10 = 1420502627 p12 = 319986975727 p15 = 360802459587989 p16 = 4907888847800929 p28 = 3948020931923766526433793457 p40 = 1069158660766613132724682792996050143417 p43 = 1026367444671582478750340776100892291660437 p93 = 269849138487532374198877021349930016361247184220695675812643691485179934127024489200988726183 p166 = 9828027147972763940840419206437324649030146749137361238797661494445753505040629992175660612970979552501716524157244914789537218745479849573839100489653483518323130703 Prime factors of N+4: 3 x p1 = 2 p1 = 7 p3 = 103 p3 = 167 p4 = 1823 p19 = 9436961405541853927 p67 = 2442634599251757965491697450488141846145584325007794753532934341089 p75 = 600199651586709792321245759194699649871225663594379892041934006939971309607 p330 = 527494574388111009719149625221156839723560610238541931586701578366691138991354382424573311558118964044710317560296491439030482498012276207040809762647265709450370220697759850439666381328859614847517666066051906937892145477551456225050086697670164991720248902858139512339587289553072087611949432974971329045183361427759949669197151 Prime factors of N+5: p500 = 12815419025806882624989107869626377225388252892348109143014972905583523571403186585248358925874769762143571465837894872353231546616266578261353426940275947134455203755973216086142017823730363315581887837636864244178669765557164623788557589071633903483494919111014070818792637091615757088898914231197548672788150791324639157778351397381132289522595508576244221687211708616979359070975986888468154087947303237066107215682723664525548690542907933932647424100409739619639031424304876201990797628540181049 Prime factors of N+6: p1 = 2 p1 = 3 2 x p1 = 5 p2 = 29 p10 = 2664932713 p14 = 64053375527497 p35 = 12950177265791031072639356885920439 p122 = 34173198741742332396716976271692371701489770814403893768924387440612462806903842625552319985422420268803792535665370032811 p317 = 38999042013591597497050209246905972783624407330496158969655866746662086214865227946049360131542356116951118562262174103446007490605710219381435049081471459942844240208575133436379007979279145337209263917682421878854593870018670730188222715372117686044826074644600648361727317428823657930132716062226843897099648501007 Prime factors of N+7: p4 = 4001 p5 = 33739 p491 = 94936245678694753421140460680692013378796538693567732159367853178703631476783480448453633397089313300647032629923781631678179232691653535690986312228338682353146884415957097940177644344733219430714565924428257796532740651918473772206920031689474586535017443888931964071673900277825984157943399173455314798320713779027598221953384155970049616308951516501889241279381306285653008566643623842165915202986674691371225006092675418281597614933590870433844042771352714591432993468524864708352370609 Prime factors of N+8: 2 x p1 = 2 p3 = 829 p17 = 26342725653094583 p28 = 1470343579963822912245711071 p33 = 187007870852441168222504281550299 p53 = 99610574090928502236269502589968168206515652266513541 p63 = 113895390854000729877269012409350207774928221694641996544761053 p88 = 4913497600365917962155071410041034922653694031638944416398503899346474340572860479110111 p217 = 9571423313604982533704828521048066928977532586908807859259654574404430464964151003988095630328625510537874340855221392615775029906318151824100974216194570356545874213060957042918269033449215300619413721816385325826807 Prime factors of N+9: p1 = 3 p2 = 61 p4 = 1019 p5 = 14243 p5 = 52883 p6 = 269749 p8 = 73255439 p10 = 1462827239 p22 = 7462473053081018587763 p37 = 1071679548735911780408902157698216943 p44 = 36445304291066947549904502068646535721336497 p90 = 484092028964579690545000872590192774860759892031294832503076409560989933338876408875527131 p125 = 50995613413068475078730791162445890545117220641170507071916755522357759495828448552610940703376080579233279139258127608797223 p147 = 438679328453087683131899883194844964328056296306402874587431956843033351755942263661835474320624975100141975738613328658670284212164843893047348661 Prime factors of N+10: p1 = 2 p2 = 31 p2 = 71 p3 = 479 p3 = 601 p4 = 2521 p5 = 10079 p5 = 11351 p34 = 5160785634557467417755921932864257 p35 = 12956653104462928747086439183606463 p43 = 2976475056377687756545777040724081181425791 p46 = 1111195533664096264407102983373638795453449159 p94 = 1411236536953344109648201030313903202292715855689198457423066135444942202534466870230539797279 p112 = 1202453802380202612900570567249140337265977386730165272588662040901323849540240433058583909567403721087774087063 p119 = 93428370649115500684161484486657066107657288857729057476468058018820851220113189554394219484855663733342770455665657679 Prime factors of N+11: p1 = 5 2 x p1 = 7 p2 = 19 p4 = 2039 p4 = 2203 p5 = 18917 p5 = 20773 p6 = 128663 p6 = 206197 p6 = 337283 p7 = 4205437 p10 = 4253763823 p14 = 10361106142511 p14 = 48219445919633 p17 = 38944767299226517 p18 = 823146395611037243 p19 = 2567947475640367903 p31 = 1980406199705343784496335146833 p46 = 4068205927282173242680543824629898847404490879 p105 = 170724646709285476974463174780574714342378046069994801064788497187739411448501668215598461179709025266247 p189 = 172234777294748971503064594510341553906830528975589697516125816488729556569330134189448072867155899116079219277251511588963969376789767369662120332326617391898100471591557566930797914101561 Prime factors of N+12: 6 x p1 = 2 2 x p1 = 3 2 x p2 = 41 4 x p2 = 59 2 x p3 = 199 2 x p4 = 3907 2 x p10 = 1067282959 2 x p59 = 90695686907824373614544935809648303855581556170634564267689 2 x p67 = 1045590928167366905609128138910595915667531392289262793343319247317 2 x p104 = 13281367071696254893447093955890080700243546027305640807619578081114123230285603623403537002718792146591 ************************************************************ Possible extension of the sequence: The ends, N+13 and N-1, were subjected to ~1700 ECM curves at B1=1e6 and ~500 curves at B1=3e6 leaving a c498 and c472.