Factorization of 16 consecutive 303-digit numbers N+k, k = 0 to 15 Discovered by David Broadhurst, December 24, 2009 ************************************************************ Formula for N: x = 9*(2^123 + 23815893) N = (x-2)*(2*x+9)*(2*x^2+3*x-8)*(2*x^2+5*x+5)*(2*x^2+7*x-3)/216 Decimal Expansion of N: 521341191752634055766740682837859368239376916178813493185322764335643216709373758128128612701680944300591301624471063359219253339521181288408703236850564896833506087086205362426450370418462121803077223769852399148761946700388138647760550294204612114194644899419256918517751457660511070513927401785031886 Decimal Log of N: 302.71712204037708897 Prime factors of N: p1 = 2 p2 = 11 3 x p2 = 19 p2 = 53 p3 = 587 p4 = 2017 p4 = 3137 p4 = 4789 p5 = 26573 p6 = 867829 p6 = 879979 p8 = 14389807 p8 = 69184163 p11 = 16896884989 p11 = 53774863013 p19 = 1879008811235662619 p19 = 4630314331114981123 p25 = 1703061512768342214442001 p25 = 6285416108194070335252907 p42 = 940368891794493990433427415734355940541317 p49 = 2155660308084192641137069464587238311002029884471 p55 = 1057504971568760490197681314870322080883653923824313019 Prime factors of N+1: p2 = 79 p3 = 373 p5 = 30071 p11 = 44889130969 p12 = 111089521507 p14 = 11413543909891 p260 = 10337212373632021003950827103556004293640854639903742277638742753568131610769794800629144953523950501478617447320393125475267744186363510228182456427674921413376814425380623653374014570683236230344322671093977476961362716705147125010103924876944835694589111947 Prime factors of N+2: 4 x p1 = 2 3 x p1 = 3 p2 = 13 p10 = 6201517039 p21 = 722871652977418705933 p23 = 95226259366865385557033 p246 = 217459848234077434264562649489721777721111733345672455574622943672448969573261577790067127344635437989207260258357322847260632414431199535054340846019467616077640521356396386610356627245871302705784377676680194824168193306276468449705025993893033 Prime factors of N+3: p1 = 7 p3 = 283 p41 = 13593308751384767231011741919574447636629 p53 = 35842040523885612575783198982737986591275558483840133 p56 = 67605347833020313440345066686572374342131956226441503511 p62 = 20080886729823073016277188560377423951121258368166359451198589 p90 = 397883307019347095022222771895670753723803337962317822028776605759120436688451605592641223 Prime factors of N+4: p1 = 2 p1 = 5 p5 = 16363 p7 = 1064771 p9 = 602762491 p10 = 3857803223 p19 = 9600711675009688319 p43 = 4121891401652115743452285161600470601530557 p45 = 577550649847139215701142830768015186506554691 p65 = 28540670536186013721303403496592128189699895086995278452570044663 p103 = 1972706081467607168690648084608964992752166863901379320256826426404872332013730102729075831896494720359 Prime factors of N+5: p1 = 3 p3 = 733 p8 = 54427969 p292 = 4355867717283690814239138332265719675131278481564726944765591604597837022494990061166338631092554240056079627375297846505205685308426729963595915903339717238859069565701816287827110622220785097769513291030085792140533750170263476560808738700714898696316211135802037623604760426400789032172261 Prime factors of N+6: 2 x p1 = 2 p4 = 7351 p299 = 17730281313856415989890514312265656653495337919290351421076138087867066273614942121076336984821144888470660509606552284016434952371146146388542485269030230473184127570609623263040755353641073384678180647866018199862669932675423025702644208073888318398675176826937046609908565421728712777646830423923 Prime factors of N+7: p303 = 521341191752634055766740682837859368239376916178813493185322764335643216709373758128128612701680944300591301624471063359219253339521181288408703236850564896833506087086205362426450370418462121803077223769852399148761946700388138647760550294204612114194644899419256918517751457660511070513927401785031893 Prime factors of N+8: p1 = 2 p1 = 3 p3 = 443 p3 = 607 p3 = 857 p6 = 423727 p26 = 14196476095207115645059099 p27 = 124287914840497266700526593 p53 = 53287310118012936318846281766919122947961842453441461 p64 = 4071628978216066685209791687855824455666093352344095034542166519 p121 = 2324395148588116224295375677316108557303920409453486416756868949486425932094333179067091889094554622384218299577607653107 Prime factors of N+9: p1 = 5 p7 = 7901893 p296 = 13195349310668571588269815418605626986834089405635168514312273384001611176192179725241245678767883703325046330656997338719196864334183752890825103221482874972706061372539601901125473868564459726373850513284662273932637323749844211956819721406114006205719183983363402124471983046606960395791929903 Prime factors of N+10: 3 x p1 = 2 p1 = 7 p2 = 29 p2 = 41 p2 = 43 p2 = 61 p3 = 157 p3 = 233 p4 = 8273 p5 = 25391 p6 = 455941 p6 = 553703 p6 = 688063 p9 = 222110477 p9 = 914371963 p21 = 102940897164342294199 p21 = 447938511803476740301 p22 = 8953727832105291034507 p73 = 1355532808022960539375434992436335168938632543775451160287781891918963871 p113 = 19675669390346017881430511708247076724587219203708522425242666618595429091642246599146686136628614963033977789953 Prime factors of N+11: 2 x p1 = 3 p2 = 11 p2 = 73 p3 = 163 p297 = 442564303215900543180133703484003297314159254685533792573455170526717054322851812628451599533176070564109284817645369875933257560495433610335392955396952037250822441650054085205742924172782639236365014774904604621525742932635998312192052718295325822469288989923825971724770571213870846046758365897 Prime factors of N+12: p1 = 2 p2 = 23 p2 = 37 p3 = 223 p4 = 1831 p6 = 250853 p9 = 330826753 p11 = 14274436901 p11 = 52218721447 p16 = 4313702133453749 p17 = 23831510979083923 p19 = 3808679457144757099 p26 = 57834966748800851592640649 p31 = 1858000355349080123334213987991 p35 = 36048812031015776277272491322687537 p58 = 1504491024202091633518873280050039551229496756100732355681 p61 = 5314606809845634340325150219291629244293823069909031599129819 Prime factors of N+13: p3 = 103 p3 = 107 p3 = 199 p3 = 641 p4 = 5381 p5 = 67759 p8 = 16130899 p12 = 207510139009 p16 = 1934210837688533 p28 = 5738714204409690362849333627 p105 = 236636308663962837331898354557647130136745440758267330709838459818270536719232425313281364673073453712907 p120 = 115681062129923291250813736671793337110288010558187313006321157423729709072227356805509070538836286319751786447973678437 Prime factors of N+14: 2 x p1 = 2 p1 = 3 2 x p1 = 5 p2 = 17 p3 = 241 p3 = 331 p5 = 11593 p5 = 17383 p8 = 31770191 p10 = 4244004487 p23 = 33990317953387929301561 p27 = 141277820922513428123364101 p33 = 111986431330001480347366428499801 p36 = 264742505384547559748476094994188527 p37 = 9570441569651394284907410834039915251 p52 = 8796710815176185373355008926006586884439593783728409 p64 = 3934776020336769949449424196119361468196862250468526544060433041 Prime factors of N+15: p2 = 13 p3 = 109 p3 = 139 p3 = 263 p5 = 12253 p7 = 1302911 p8 = 36612473 p13 = 1486237116289 p13 = 7011811113257 p14 = 29018294822527 p14 = 64273240002817 p17 = 87129383658067867 p18 = 514254020143501811 p23 = 19264685041836412340651 p56 = 66618110634310045656169769862925578885552757785291480971 p113 = 15405593623077897136937309672810255118897394172922322742094095265807345632044695132863936643470958566885085078629 ************************************************************ Extending the sequence requires factorization of one of the following: c221 = (N-1)/(3*5*107614765903*707599683401162722538492715491*5791987309280956264759051605269253758123) c258 = (N+16)/(2*59*347*509*28099*71147*63442396558253*535991020920557) which are unlikely to have a prime divisor less than 10^35.