Factorization of 12 consecutive 521-digit numbers N+k, k = 0 to 11 Discovered by David Broadhurst, December 6, 2009 ************************************************************ Formula for N: x = 2^97 + 51514439 y = (x^3 - 13*x - 4)^2/64 N = y*(2*y - 5)^2 - 9 Decimal Expansion of N: 60519105271400383280724193235711474399163050011215808808691950551771627685927886559527104698631809236563584713577453647347175576098532997674540872219848899523400890175582131681653066871403094664565957432716762301807576834919872820081424456905926749274581639355923484329378225200559149737266163109986220887239430918566514269231849716956863254855520332593640840090021906611350048326811825202121189631687490993477900278997064436147223271491590399551705880306681861367421014609922625216326420910670609814157472523195335753952 Decimal Log of N: 520.78189249868746772 Prime factors of N: 5 x p1 = 2 2 x p1 = 3 p1 = 7 p2 = 13 p2 = 17 p3 = 887 p4 = 2437 p5 = 13339 p5 = 30089 p5 = 49787 p6 = 295847 p6 = 699089 p9 = 631106473 p11 = 11758433179 p11 = 73686673513 p13 = 7997685879317 p18 = 360477880611768553 p18 = 518091460634382709 p23 = 15050964904693165182299 p23 = 17917308789525613965551 p31 = 1171398140920239420717280663859 p46 = 5728960191355159203801240850367037084832744087 p62 = 92392243208124716402416696390826938854637442681347332860841289 p105 = 121236226055886382351608060819534752703322469026614011168245516475418263183019798929549185071705234086761 p120 = 918349743607285428786924077404455720960195812218071791254827049991798749780311888599251213291805037642077080458866120179 Prime factors of N+1: p3 = 257 p8 = 23296519 p10 = 1242096601 p13 = 3048355236167 p17 = 67973327977858879 p18 = 135996879951639809 p111 = 606635802341662435132175179349293714549272051430455678036875643598915349310782211721973275723042298531808064519 p345 = 476049241189471357760864128686221766350096529085909228324245109554467660977093847926313967394428260917981014913568279257549965254742677328761804745011575095142481178743211265579508624993461874944839192347838538543548727181449071950041770613073355148416139112738412832441110967186585544024820400112942605725121208812404616109968539147733573432097 Prime factors of N+2: p1 = 2 p4 = 6857 p517 = 4412943362359660440478649061959419162838198192446828701231730388783114166977387090529904090610457141356539646607660321375760214094978343129250464650710872066749372187223430923264770808764991589949391675128829101779756222467542133592053701101496773317382356668800020732782428554802329716878092687034141817649076193566174294095949374140065863705375552908971914838123224924263529847368515765066442294858355767352916747775781277245677648497272159803974469907151951390361748185060713520222139485975689792486325836604589161 Prime factors of N+3: p1 = 3 p1 = 5 p2 = 43 p3 = 863 p4 = 1747 p4 = 3581 p5 = 21019 p6 = 179951 p169 = 2665988320602598119712037148890597589508913576108560151252151200435666518468042695203986763274362208546291758588761326187257881915092235910607165312317946140409466980679 p331 = 1723456019694535985024052831278769982409850693731036183019401249854005649027058715170650019790949774813382424909094653917094387303200560837954607615294370400024910950769859150808458713206581975301028261726486069902159812186060487965885791282409908342513319078236674752345844445787295726398459700590072492402991653840119101403324069 Prime factors of N+4: 2 x p1 = 2 p2 = 59 p5 = 22091 p5 = 35999 p510 = 322459133920738719756379183082614359056868778245850768245023823013839191741970939687061103438714083836188959964585016323187874208238891491852611333081652500839247896404263392467837072194949382142286276046932466506539598761625247632987585625810478977873246638233056415567226543515573503569047983938340923349002601800362805595378175748488957506319224702588948731198922660800735914986293556480183138801983977067287444288997752400559933169613025620686694739072055976737158292295388315717346384419219357830330561519 Prime factors of N+5: p2 = 11 p2 = 29 p3 = 101 p3 = 571 p4 = 3691 p4 = 5419 p5 = 65203 p7 = 1295389 p7 = 2920079 p9 = 138203671 p10 = 6079815041 p13 = 6807805279087 p16 = 4214598470855783 p60 = 252775275481837740704983146954700963713586730153850922755617 p147 = 576545967736398377880005536912745355844295932795061127507304915071378800145932900134220140776910446588573801936420449624688244935867703659255654133 p237 = 189791581562086004182786053911463893927100346909312552275180515756725069187771439438525215323275001305235375297858441337994282392220999296010244121821682564249311868864678056128116961654457526895927237549982385731864650276028768837996759 Prime factors of N+6: p1 = 2 p1 = 3 p11 = 54881331191 p13 = 3549757051333 p15 = 112712844601859 p18 = 345742433728700149 p53 = 61347189073381493207918536815318678415683328792843643 p80 = 14608757911619482325513417398922661995740823683847143154630043302377343949721739 p82 = 2384710061042407663350652645312256059993060891913655975778396102465978591303335049 p110 = 48974153515499894645234965703965797304135752880534893557058100045319995640327414406010484894887092334290124777 p143 = 12693500454889611206532256075665637940212171162010189388440729662267877275515941250815914080365061254677361638056776706610356922013384369276221 Prime factors of N+7: p1 = 7 p2 = 23 p2 = 47 p4 = 1823 p4 = 2273 p4 = 2447 p4 = 2801 p16 = 5517624133009231 p18 = 298951889844775159 p25 = 1583866351585911133838401 p35 = 62186682346004723275580488729752311 p37 = 1468062138781954877605806181091162927 p38 = 17659000145579387114655129432669393871 p74 = 88970230145880244982401312177602815659065896037765606484677123557913151049 p94 = 6906100861906711042294651808315507604303950918775090357302270975396916428842528615127185377473 p171 = 108812319169865611113499765242170244114701137148935942205033082816725856614419925622882473381377372631922944203113004898243458798930001960412143970055880623982022558497599 Prime factors of N+8: 3 x p1 = 2 p1 = 5 p3 = 109 p6 = 106877 p6 = 271489 p6 = 548347 p7 = 1889029 p15 = 690425747753501 p19 = 7128686386107336359 p32 = 61723043861290773050031488327281 p36 = 327431229614131117772785108403020409 p80 = 42690181228630617580877485609562883338265519449446054930706854035269082610743203 p86 = 32324931823745846776555676474953076385481167075445075674853087885869075715823923787917 p229 = 3364463967918569877412428970831620168291007106120572015124936370187375287372651445665862816100243713642209184736833502743289161172062635744244868288210173829656917423066932066331645933242315304125288886809659520629140738792514459 Prime factors of N+9: 4 x p1 = 3 2 x p2 = 31 2 x p3 = 271 2 x p4 = 1213 2 x p30 = 290717660692056178627380230059 2 x p56 = 55183042568818223466137174736479621212999091619327075187 2 x p64 = 3286538828791738865769644759971695914014392529618376267704168559 2 x p104 = 50874028544198389096448371570878800888606150471855795807934093984638655897441174063482462519436790916281 Prime factors of N+10: p1 = 2 p10 = 2099914577 p24 = 969572514811180033060969 p488 = 14862113683172346660679035738827879936503205566178442333744661485784153322240576064036626055026150139718118709890408244935806017820650482668730245340130377513535820591485115764108118201599620990954914036508402294340227732719138362867512750271042304670949378877240075766012145445424731478844676628946664186449929860837596339376487745990548334471859062988992666432445211023935619470361596353906743734860305940650004777588660667165450359950515441020001397969972969333797576335749250360580637 Prime factors of N+11: p3 = 131 p6 = 307147 p25 = 5268867920473195353617881 p489 = 285468118992391829648419322477201209100810958885216242326353703394434170611133761961110166710562344820474223372358500167065188638929809981440647818911235780941932781634678098072066413996074788676381917030391682632136783549620672727830392653519569465479500629675530061165308219325805916861198591256748398089833090215976425658181319056661398796782824490235834173491312585430636440166063586497766854038854435969647736710828032362938950467684740417242575279044170005904660447932181077900851139 ************************************************************ Extending the sequence requires factorization of one of the following: c500 = (N - 1)/(227*4373*1190082265745281) c474 = (N + 12)/(2^2*3*61*631*1783*88903*46906094056543*45467031493033267549) which are unlikely to have a prime divisor less than 10^35.