Factorization of 21 consecutive 303-digit numbers N+k, k = -1 to 19 Known 20 extended by Serge Batalov, March 7, 2017, who factored N-1 ************************************************************ Prime factors of N-1: 2 x p1 = 2 p2 = 23 p2 = 41 p6 = 290399 p41 = 82049473563797795110840675341962134165303 p253 = 5800687119043966076458093431403599342066486860130593363750638259379043892237928868657375528720659036199535857003362322259881288749204326339939518541384047634456777775176259082126086907366473591808484877382252182189655036384810428841744360051529320492751 ************************************************************ ************************************************************ ************************************************************ Factorization of 20 consecutive 303-digit numbers N+k, k = 0 to 19 Discovered by David Broadhurst, January 5, 2010 ************************************************************ Formula for N: x = 9*(2^123 + 440959466) 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 - 1 Decimal Expansion of N: 521341191752634055766740683001468717552962752107827574881859021123436252765301243159669343194401835687604516077752513002479438195121716831217613937259767161462862578867158043359385874909266375586474588655395395254097607326657718327239862747853999739627078767683705549273926025366885135785195664000024485 Decimal Log of N: 302.71712204037708897 Prime factors of N: p1 = 5 p2 = 97 p6 = 107209 p12 = 154276061227 p23 = 11107151800249381618681 p262 = 5851238583758916615765566664938759221914517482036677801260817175226522197257896081251054900329309745615253559613022880559754224847518500164789886764872439173332971995989774307374793223443183819136457790269706712003430042276065080978076113753501552115503346126147 Prime factors of N+1: p1 = 2 p1 = 3 p2 = 11 2 x p2 = 19 p2 = 37 p3 = 173 p3 = 257 p3 = 397 p4 = 1667 p4 = 6793 p4 = 7177 p6 = 241667 p6 = 275941 p7 = 1331959 p8 = 48667933 p10 = 1245724411 p11 = 11306983447 p11 = 27603788687 p13 = 8134278291367 p19 = 5165989183763144921 p22 = 3084279880378392542453 p25 = 1579286578200492977511637 p28 = 3600878459806435782033917761 p31 = 5453311391841668853138230012959 p40 = 1675062260592055659167406119590317872017 p50 = 36430703529892257187155843659410127687966033308507 Prime factors of N+2: p2 = 43 p4 = 7537 p5 = 23671 p47 = 21736936592318538144041717556296066161071377743 p247 = 3126368206947347642885954108490113083654384155091239337976492685352052559461513012446362067395433732602992321830689561676282878843217253440806806770541464789650078270282602205694676872169540694849055647415484812285175472945223265030622798602943469 Prime factors of N+3: 3 x p1 = 2 p2 = 13 p5 = 17783 p7 = 3064421 p19 = 1562164195379629321 p23 = 31667585951950278223717 p250 = 1859488963973290558098920281484486831828114843650414031044445103341253103036254717068902074154073506256584794962530768550867337124204930737691139914691991739935391174811576387856970859646318643507064822503900088775096604887036878822934681420000326047 Prime factors of N+4: 6 x p1 = 3 p1 = 7 p2 = 73 p3 = 241 p3 = 313 p11 = 16938308677 p23 = 11725129289235313846261 p117 = 551097019193377430838204980605663349327507824752366585931755195658524482180110045241718300101636424124370428915912267 p144 = 169510453807939551386515480696164049359872503742428678826560213175208012272611260611895332886977128659950207637119728690462565349164507961908093 Prime factors of N+5: p1 = 2 p1 = 5 p6 = 440171 p7 = 1320019 p7 = 3046597 p7 = 3531691 p7 = 3551447 p10 = 6139570597 p14 = 15552842603983 p26 = 49243669754870476532890433 p32 = 13331764107684919178425154908691 p91 = 1030895553349963942294198660492583834178547875186417835264150660415576398420148912432564039 p101 = 36334309689753682691151892731982943177919055671679546786407538020571056863845970314681374747339502487 Prime factors of N+6: p6 = 127873 p298 = 4077023232055508635652097651587658986282974139246186254188601355434190585700665841574604046158312041538123889153711205668745068897435086618892291079897767014638450484990248475904889029812911056958658893240913994776830193447074193357783603636842802934373001084542519134406215740358677248404242209067 Prime factors of N+7: 2 x p1 = 2 p1 = 3 p2 = 17 p5 = 51941 p296 = 49201865139654500125400641508546906874444151764561258879499686967928189711224126767481405485560524336209949003012138678696854594364582291070223901974352419606452284932938432346446805114595177521032969596291134554071494328091122084525755537472003466567749642003663427817792324074231012467123865653 Prime factors of N+8: p303 = 521341191752634055766740683001468717552962752107827574881859021123436252765301243159669343194401835687604516077752513002479438195121716831217613937259767161462862578867158043359385874909266375586474588655395395254097607326657718327239862747853999739627078767683705549273926025366885135785195664000024493 Prime factors of N+9: p1 = 2 p3 = 131 p3 = 727 p5 = 86243 p8 = 17122751 p10 = 3189848293 p21 = 290648523889460623331 p25 = 1613128302900970825296953 p26 = 11553698066355951684885457 p29 = 18986543340758676314627275109 p63 = 126077688009562744026545540726968896931740377047861408566524231 p116 = 44810183783314201027239828842553372944626352164658358315135360688771658746684573752767621299653443846747621522587211 Prime factors of N+10: p1 = 3 p1 = 5 p5 = 11813 p5 = 41969 p293 = 70103863974464461119989349223316404124331129492398090447919700275515160179779201875474209530217684618238490004162407476032407212043201622059052147736705396174485178933468863309273555696376547534936441235609740568499151291106727412323319923132997926446644990078623628582956830377779611403460189 Prime factors of N+11: 4 x p1 = 2 p1 = 7 p3 = 379 p3 = 509 p4 = 2389 p4 = 3191 p6 = 106033 p14 = 25914136009147 p18 = 649654739891712887 p37 = 5316911983139663491615228241341858037 p47 = 22636489484054749883774118474691065922693126843 p71 = 20231989370390773818346874018254116351213333923953506803692152361326059 p99 = 728176302894991225616484829284944354515633811471140098808633943346867076735120622293318348606181849 Prime factors of N+12: p2 = 11 p4 = 6173 p5 = 10061 p14 = 11112245169497 p281 = 68673647445480663876132192066598178097056936468278252448160398433832395330017452912585435148534835427727884799463222139005455417942226356405272972590219220685112620825751899960786156949293571777948321228992871630141437375281198476344229171679775914351118714232245969492798874743747 Prime factors of N+13: p1 = 2 2 x p1 = 3 p3 = 269 p4 = 1171 p7 = 4325621 p7 = 6796177 p9 = 160301111 p9 = 338896009 p16 = 4138918069464307 p24 = 896680516487134293256679 p41 = 14846562401327463086255266699568893125359 p43 = 1005817326665491207327191295642285234440893 p71 = 58154693721638128149992947562442556396617950567732606738974046387589893 p74 = 17863636422086177970038371123489469010861979292462479294726319804133349471 Prime factors of N+14: p2 = 83 p9 = 149347621 p11 = 46456996231 p22 = 1671384875800483171177 p45 = 405385145332468280102014913544468396259654121 p46 = 8753523679366571750412435727703415486312790187 p66 = 143339148158065331811257236557649266309397086148824671306092988641 p106 = 1064884447075749001498555395438379621888577989401739835922873815837015352108610825142741971795910254352177 Prime factors of N+15: 2 x p1 = 2 3 x p1 = 5 p2 = 31 p3 = 101 p3 = 137 p3 = 179 p3 = 307 p3 = 349 p4 = 1583 p5 = 49727 p6 = 177979 p9 = 384572099 p12 = 709173334169 p15 = 122920492355971 p17 = 30919838615317979 p23 = 34520056429125747938483 p59 = 47071576767018600338879876119677058070309046103314063271841 p66 = 112359395095090934990265032774898660143727230975050439866745290781 p77 = 47803348216449732017392943140126258266740966895918692804251461359100663919217 Prime factors of N+16: p1 = 3 p2 = 13 p3 = 311 p3 = 317 p5 = 15797 p10 = 6732700319 p11 = 38571501049 p13 = 4240278661351 p18 = 561726160697195909 p19 = 4463585111564238587 p24 = 878979931474507754736851 p26 = 45140625576726341976777283 p48 = 281826289409499606714162636704035013624159008349 p51 = 145149491556361894697240213129050729078969389513737 p76 = 1915406321403437148858484605969202267470095346191222618292208928812911859343 Prime factors of N+17: p1 = 2 p2 = 47 p3 = 607 p298 = 9137039359119388267495192313110671904955707387357208014333818590266680443851891814638952350141992984114489047596349556634993133217457969631210591630617392152947221754480669553075570032410290854682508827077629697046822659866411692089450431978933712005802495139747371959653791324036684352504393143819 Prime factors of N+18: p1 = 7 p30 = 973801266503718909214378669723 p272 = 76481018940259023433666995264261567354825357307107950538274685213921068668779446131571052396950459463753208521185278478324026464373467974640475117855364309560942472261586841736167426794275147349996728494231798754597117544468144601637295164782147689564582564931687544613123 Prime factors of N+19: 3 x p1 = 2 p1 = 3 p2 = 29 p12 = 295275230279 p289 = 2536797399898501074455228038768410268362932727905939830410011070240028237256382759956497301361001685799278784407860359672198748035193617818560119096442455437172898581867336409633950074488225423246239721055343084234577456951250493502741147200179012001530213390702428228161819432719817982231 ************************************************************ Extending the sequence requires factorization of one of the following: c294 = (N-1)/(2^2*23*41*290399) c253 = (N+20)/(5*19*205889393657*301133059382193691*57657649694534487023) which are unlikely to have a prime divisor less than 10^40.