2·103020-1 was a known prime number. The following proof was generated by Primality proving program based on Pocklington's theorem on September 4, 2003.
input
19999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 9999999999999999999999999999999999999999999999999999999999999 11 41 101 271 907 1511 3541 9091 15101 22651 27961 51341 92111 2307281 2455261 20481641 62459641 124254881 181288487 429360649 116848737701 153653579173 311198063831 352415012896091 5709307758717571 102457179041535481 12331942624071394811 36117943403747426761 168782302910214142741 29877981480806017715604631 2646756758131654397378192081701748003041997 28656552837574546515657117360191971140211910788651135283 3332366532260724652058380106533605034666939213869789759449 999882306178377345161889846655198592766347086896770989636138585751423 14566312907843031759294690462628712751924775577830480082972863481356970368374590\ 09760568165838021197932780842262487066607002829530360463745583774491737217277822\ 71571817561233543303249152299316495601747752344551443281621150615663216827792895\ 11738981674621078415475510603090310250565861705288578763056615319206750386812201\ 01808015883102234308891744099243060004258137332482581748971973482755492121397939\ 67559963577127431921398927624119559898628967968934152391850673474805013544123283\ 33487325067420269240933678428616588450990603704599633210385413199013705671355363\ 4419983194861986470725514079191581 0 0
output
Primality proving program based on Pocklington's theorem powered by GMP 4.1.2 version 0.2.1 by M.Kamada n=199999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 99999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999 f[0]=11 f[1]=41 f[2]=101 f[3]=271 f[4]=907 f[5]=1511 f[6]=3541 f[7]=9091 f[8]=15101 f[9]=22651 f[10]=27961 f[11]=51341 f[12]=92111 f[13]=2307281 f[14]=2455261 f[15]=20481641 f[16]=62459641 f[17]=124254881 f[18]=181288487 f[19]=429360649 f[20]=116848737701 f[21]=153653579173 f[22]=311198063831 f[23]=352415012896091 f[24]=5709307758717571 f[25]=102457179041535481 f[26]=12331942624071394811 f[27]=36117943403747426761 f[28]=168782302910214142741 f[29]=29877981480806017715604631 f[30]=2646756758131654397378192081701748003041997 f[31]=28656552837574546515657117360191971140211910788651135283 f[32]=3332366532260724652058380106533605034666939213869789759449 f[33]=999882306178377345161889846655198592766347086896770989636138585751423 f[34]=14566312907843031759294690462628712751924775577830480082972863481356970368\ 37459009760568165838021197932780842262487066607002829530360463745583774491737217\ 27782271571817561233543303249152299316495601747752344551443281621150615663216827\ 79289511738981674621078415475510603090310250565861705288578763056615319206750386\ 81220101808015883102234308891744099243060004258137332482581748971973482755492121\ 39793967559963577127431921398927624119559898628967968934152391850673474805013544\ 12328333487325067420269240933678428616588450990603704599633210385413199013705671\ 3553634419983194861986470725514079191581 f[35]=2 prime factor check f[0] is a definitely prime factor of n-1 f[1] is a definitely prime factor of n-1 f[2] is a definitely prime factor of n-1 f[3] is a definitely prime factor of n-1 f[4] is a definitely prime factor of n-1 f[5] is a definitely prime factor of n-1 f[6] is a definitely prime factor of n-1 f[7] is a definitely prime factor of n-1 f[8] is a definitely prime factor of n-1 f[9] is a definitely prime factor of n-1 f[10] is a definitely prime factor of n-1 f[11] is a definitely prime factor of n-1 f[12] is a definitely prime factor of n-1 f[13] is a probably prime factor of n-1 f[14] is a probably prime factor of n-1 f[15] is a probably prime factor of n-1 f[16] is a probably prime factor of n-1 f[17] is a probably prime factor of n-1 f[18] is a probably prime factor of n-1 f[19] is a probably prime factor of n-1 f[20] is a probably prime factor of n-1 f[21] is a probably prime factor of n-1 f[22] is a probably prime factor of n-1 f[23] is a probably prime factor of n-1 f[24] is a probably prime factor of n-1 f[25] is a probably prime factor of n-1 f[26] is a probably prime factor of n-1 f[27] is a probably prime factor of n-1 f[28] is a probably prime factor of n-1 f[29] is a probably prime factor of n-1 f[30] is a probably prime factor of n-1 f[31] is a probably prime factor of n-1 f[32] is a probably prime factor of n-1 f[33] is a probably prime factor of n-1 f[34] is a probably prime factor of n-1 f[35] is a definitely prime factor of n-1 F=f[0]*f[1]*f[2]*f[3]*f[4]*f[5]*f[6]*f[7]*f[8]*f[9]*f[10]*f[11]*f[12]*f[13]*f[14\ ]*f[15]*f[16]*f[17]*f[18]*f[19]*f[20]*f[21]*f[22]*f[23]*f[24]*f[25]*f[26]*f[27]*\ f[28]*f[29]*f[30]*f[31]*f[32]*f[33]*f[34]*f[35] n-1=F*R F=328853317866847139922365636595893146176225859284846910914338677281295187648908\ 71322226622794306837578463510825730646110136123627057808662791341530906498982223\ 94905707620396731885593839064405703939410350373431395114949899374628234585141530\ 48440712180645783230629244807237339608506459186155216763542682302915462475214627\ 93522852335915216655217615552792424214316546631885729109772274281822974015293003\ 58652483118381295763731174318306889682638107401591614605307396134597913674089148\ 13725484444635685754097226929970812839654534547495459836777044150082735769918959\ 66240937932969627070864464121687827426036790677838366043606646845612602555880455\ 67195847979082580493432043985546583215500963332731847912567509961171906256374199\ 47016230530335352216827497802222953853597280865454982348524888562362320465141658\ 62047978252153347600951121633431861133636980921065697979178763586853728153485888\ 76184632247712511365976082571513193858065809375490773274564001869658661643957039\ 36892626260841948042633409227778467399751973632074183007111091986980145832862558\ 474271590102100805976686546064455833289742 R=608173885236517796800120208926018052205718934134998556038142323323209705910461\ 03229998920070089027904120224798717950537770837780133721250353637414131332077810\ 75989128331356470653058307563193794244385957622111428456723906610578967485976936\ 77812761029227634795138866996211196253244632086928187097132879821810216973324071\ 25603031017603215170149266282010033771820891174539482777090215247389853805081422\ 35022885105259469283380370231865562138049825748804498891348890709543738798886111\ 83850856529801744443976916235524721695980621734782100479734251420786696335841117\ 06924436669239992081320673788949774222076731977237037999633653499011215591076938\ 25319043977355049661360572589594115079424999159496650119024122927572028334336385\ 66222244017777576703819349417305965813200728850520946732791855368738728895370996\ 15785346267234124050506827904883311309053652082779512933076946127520928071263074\ 76805337538870797777124296469668844816869180358770580083719368198773107466743769\ 66406515214085133803388203134666723033927813129023393393161334173530581537138479\ 46695493478929322477211893222365482157703419349783568560995216857680832053345408\ 08779440572703612019335219753949709335732054441200440348543267993154831746635817\ 04712882655626818359288207809441810274186000043932795963041054587811339035269774\ 30872872871556630851051921445915798742720937478071038529757912824647399159018144\ 45755956982812224072721524124706271982778955015764937016384270729948667151577372\ 82771536351357442669269584844978943908504582768630485712077512264384796481032273\ 71543126547345000635611099330179872645615285895917699743582265832189646205931278\ 18150696222987305047580779408299446053865699417738592746615072805535611608755193\ 76124765960812804424898797529205237441846799809757972593315467554998119406386684\ 23318558085548598822793388497065512037175862678653327562729655059651315678053140\ 76582182892100380748071928851007711058884845907629578077115336119883829242642892\ 82686938051554384720969 F is not greater than R main proof 2^(n-1)=1 (mod n) gcd(2^((n-1)/f[0])-1,n)=1 gcd(2^((n-1)/f[1])-1,n)=1 gcd(2^((n-1)/f[2])-1,n)=1 gcd(2^((n-1)/f[3])-1,n)=1 gcd(2^((n-1)/f[4])-1,n)=1 gcd(2^((n-1)/f[5])-1,n)=1 gcd(2^((n-1)/f[6])-1,n)=1 gcd(2^((n-1)/f[7])-1,n)=1 gcd(2^((n-1)/f[8])-1,n)=1 gcd(2^((n-1)/f[9])-1,n)=1 gcd(2^((n-1)/f[10])-1,n)=1 gcd(2^((n-1)/f[11])-1,n)=1 gcd(2^((n-1)/f[12])-1,n)=1 gcd(2^((n-1)/f[13])-1,n)=1 gcd(2^((n-1)/f[14])-1,n)=1 gcd(2^((n-1)/f[15])-1,n)=1 gcd(2^((n-1)/f[16])-1,n)=1 gcd(2^((n-1)/f[17])-1,n)=1 gcd(2^((n-1)/f[18])-1,n)=1 gcd(2^((n-1)/f[19])-1,n)=1 gcd(2^((n-1)/f[20])-1,n)=1 gcd(2^((n-1)/f[21])-1,n)=1 gcd(2^((n-1)/f[22])-1,n)=1 gcd(2^((n-1)/f[23])-1,n)=1 gcd(2^((n-1)/f[24])-1,n)=1 gcd(2^((n-1)/f[25])-1,n)=1 gcd(2^((n-1)/f[26])-1,n)=1 gcd(2^((n-1)/f[27])-1,n)=1 gcd(2^((n-1)/f[28])-1,n)=1 gcd(2^((n-1)/f[29])-1,n)=1 gcd(2^((n-1)/f[30])-1,n)=1 gcd(2^((n-1)/f[31])-1,n)=1 gcd(2^((n-1)/f[32])-1,n)=1 gcd(2^((n-1)/f[33])-1,n)=1 gcd(2^((n-1)/f[34])-1,n)=1 gcd(2^((n-1)/f[35])-1,n)=n 3^(n-1)=1 (mod n) gcd(3^((n-1)/f[35])-1,n)=1 R=2Fs+r, 1<=r<2F s=924688686709202799397700050155697765302111972603790780817678293853983075826517\ 01464894232231905678245892062105420543489224524136052048551353974147754533867887\ 78207506209022658382553736222707365864497488391139280748870607150951923786563072\ 07265161798599039322054707828434498999863917995587523371551783237940901219850783\ 55232869830048044134907220845151189077146154128481617650936030063666130158385284\ 13121092379708813041278968593544286263296086254127865371454047799301899673176493\ 66823510637379721792181516777792945747624918364157919533844854311106105604430612\ 42035301961984296930897216073652688650084885510673356162723138881015689918679781\ 28709457171933027491125000534726131875863451400191876806059537134852271499251693\ 00830672784267413907838509802387506189436327450041307860590348726656177516653297\ 892429942959823146510524180706545515790173021378355462976199593 r=260453520286410954689134425499126426107653040327387786774329330776753822793082\ 33164956300592782943964167487133545311455120145061044429991803172300718123360563\ 79361515899100672664895233026407831949834864988156380058927555292568297521515282\ 35802860507087303118338531540431042500999444733810381931164536459042696645045086\ 14036398099552835068342509696166141709274123170600776384047429943839011530904925\ 83883763760550493020558677203570141448905004784671616006582409077672648150935639\ 18467465323173862614231523163456871815573657522106811285200845344147158592157214\ 33056311907503426266153812393528048528714631784683181952754188143914655442161102\ 90000026828756610275074616316024795033922817296641205576327624083669989543351826\ 43349154130862122240819807578774608665493683689343438408113871455982193532755850\ 15504051572543223126233581058193748228239363097793112668553232107568581873648143\ 21639439260981089094627848161213336535453791461641301227212516006404263237855007\ 72331341638793773607439018077598420318242462082506049820678975772594714634336982\ 798314649477118995702257237062102401770957 let D be discriminant of (mF+1)(2F^2+(r-m)F+1)>=N+1 D=F^2((F(2F+r))^2-4(N-F(2F+r)))=985929352738014906663831271165765283783979933917\ 13135652441572727386023116695788106361837866045328848657548836362111025610672361\ 64514414004172091023342424444926579787042101405261995990562346353259126977834624\ 69235429641296699656064100647856880911287134437456177899154283417074435855635859\ 29377368907868584606137882490078558610346800856788733810991477932524127872345657\ 61317126665654894292033623785512645536846673006996161328439130221056382020951596\ 75581892289621678703441369194333886870678823002925402325984395951080959637487579\ 24225124372619503344616612713871919033673076091573993190783528482933526874026967\ 89083616675040321798997706854109325559382060924738071708325788922464455962265350\ 19658023938632628202518291928322184152006241756002857777333343917741127083962567\ 85533384737124357008152268560926595391959367918755489887382859014855241878277747\ 27041131379582578688671363532672841526482413253851655519256279107552173610342095\ 73545628903908511782329181557947947404584367195048931518427612623084856353497419\ 06055432502372795888323183944551115484238348265334290945625431716740524461934274\ 79040441294622171932929297312648909194237129319452110363756736927215696318199662\ 24583208480263702386002430299301135338475644649210129800103593411068721670244283\ 17459220409142153876807478019160944806811583989112259429673952420804645395181139\ 29830375932727672481789989275515120214276504825070532245352734564024963059322231\ 94321396628469206197899663282961679863754186630812682410351768932353750453774306\ 26421248035885978394391676300058040754798503719340764885996405289282944380012760\ 26194144602932270338676442646429966981614596259317921688526248658496577456807804\ 97945229719870708037317212333711414835560351624806161310869331697180054230412179\ 22545199763036637639132578219130882557625025405219439501829839089437106092149994\ 27335753103388150223411847162473071352057900805561213363947543131596622178721145\ 85885831752509784742209197266226923713271189311506843002007765970383485424576984\ 31510484483963475943238112169243200931674101987232733666780808695488979055430420\ 03474883496785743116867083854802650539573465795227324569413624229237501315365096\ 13578603877392219445494177974896749269907723894230489632032891776437946753293699\ 15290188380877185776288771573169246761427196610170918467681311712575073920751970\ 49428157886408494464290075072468228336601053490345174581156959261660777068666714\ 61247536706883798865987897256027981736688868938035941593077818682825296576458327\ 44354277620714188633623143727601351156678872741415833739591605136414246760821925\ 04745318105435624802042868110034975881858306626729014508469178830170435671244224\ 78816556295209962638193090662972818515869028695028063104568254104914333342791942\ 25372408681276747459451984165836344748940943140716742535394107018232477053827297\ 93233645837840745232690306074034469297662086485998376067588564357766549167107144\ 55940985003879040913816310704611410092421094928539092316399381839213996528612151\ 54123752734263575381451845914607195600236877983370299208009989718976703281108311\ 81316862835467824315085877985696537165793679830334012054515898320991499862196439\ 59521683300542234650541585556391150506032992256525834775005614736940811758794355\ 03132165310331636179608110993843152509743090028082609937549807522737837100238134\ 70828914899530241054674585417528497086094217913881640580621426261186247668419118\ 57241233167728324594841320830441114650315313503262471464484289395439860222717766\ 66021342321030412647460719755763976476059667380487189699454513256455190099841506\ 46908078780303188374386006502894931591026506299943008340472644190968613710813655\ 01089099669023508517123047933864324045931622879026268928932784772438515940847079\ 22412577447472410121145129480051176644120517138985649417323328987811239877758401\ 99359678784397826032939751859256905598100342808541855151566188430909406858213333\ 92976254902522661942680310176105111486945373266377624765566421655756181664606218\ 28188903680140214942960176670196101233060419654002021353867077361526364160876831\ 57464669858556635750378550236155704684428566821873203710456467445393288850370440\ 59913442911336676656571174041190781274473705199500826288851153059294120559580316\ 24750562052190880252687391026495131365024110984632908787167718749910930139479746\ 57175728905959099290903219092731189981306761043551767164348428939844010711387428\ 06888994026064916249257086800977577785430697913596733059389614277067606845083392\ 15320243955104591088810832519309912208616229131846337636928136722847750114526001\ 13409647825762983196380547711724253761514178698030412799124458223033259111083148\ 05879736969967059677926875445853437246509435320862968187903230911365564235049641\ 13918126150460641738484087135256509588618741980537750144740401260909757074018733\ 34806350774024323569877907382520256945303778487105507742207567180617018199118225\ 82074470659247348275647429398040336659041627461438792152172292740842514225886415\ 66417255585691041200629294635721104852186251107208862524758634309359912791847139\ 68724282411276603723319851320896905702126978480023161777019815435762571590588817\ 34802954871597761481186198721843504085569869911150897179625861204573542722121502\ 48089336807779963245483091084435162904546234385977305904469495861582811403282563\ 38930201356011803221428567675850632634300720620692158327905284091053217328051162\ 40806708388725048444403757997146673342850443320905731343144760841638006263102869\ 57994496719354198185531642948292717061895923972616093086877100632928658426066292\ 21863498847551832372110507645806996115083630124801698042136368937600820768555999\ 04457640669289337153602147907980967021094728002465003061335764036744989710285778\ 55509211579942766997311603781646879990529422945289903156648292398685572189314849\ 59965039629165213560841741898041251643902196089113851887163907446085437417223024\ 01637416062404786343010149348222379597621945655312296395191394505543438128473851\ 56665883056585510151697656602555092081223346236328292359447517188827248764627214\ 67860545035577415780283150784951879553054859558183705655102829749258647692972909\ 19554428395012727252824615507006826755845092584748993248806978474318118495081666\ 17036844338943167316616170379310655646958213119312631294345905149420664264271516\ 38675570614542147469901608309748150113298427426789342079595030282043004427330886\ 98776013933998922389212633767629281911446166186669429085741274223945691996726499\ 74435880630315779204654663356417498276810475103338801148309480922736060471341690\ 07425553327347840635500520613375854127958019518044961737404664256270339819603270\ 699652864234807268320171147264 m=max(1,ceil((F^2(2F+r)-isqrt(D))/2F^2))=1 (mF+1)(2F^2+(r-m)F+1)=9929397528239137363013931515452327628757521558154102331417\ 99871744510649665001263572555764412495603056805114516016396547273951413528817886\ 85696775594306832074946795992417548641114193207427328580742365873895721396102218\ 42719314107141917443778052120955247824117842942686471806559645127797349125108405\ 40589793674581906201049391462514443773940985751971250061189724030424898550644567\ 63616219672480528309263167751165425592707137551065252766494561800880473276474325\ 21034585165460999557101473853644400063513336593318107424773476471738964667441293\ 20903601273152482307241452982956290573329328138781729855422930840637177571702703\ 83888682532890022709087217013022583205508324538541380202638645414453969872933016\ 59959194310212214965883212232944908858204810852533856541316745942219700636587307\ 77015327123343309000080601493493957010789407016621103514012988030080503799319003\ 59718718720543506095436733531045346348783152956970430280649968487148478478315510\ 50134213746705950579623846845092680165513592767895743329850217181508792773825454\ 75875267527283939606876428400890124469482375437373483290847768749712074459694978\ 41486728988284588463604054785614778473173963133742397480814223913950622966701450\ 05283043872982412664740972350258477363921614818466558134383482296119499073860670\ 92272020777051590243358815717855732075947450169465939653656689666636202542715824\ 03611323279317288219740362359812288116762348100189488248755063301968374174499047\ 20258453934114273745604764685318844376568915279975374194422619928903372263321900\ 26184100245337353103860592649298190382190407085968519003453555101156638643817242\ 67094965986320810296571986949594640787519295157181216504567810255003401403971486\ 79033010464663856816439207271923337674838813690040078810823002753988422211677354\ 91219902746635978659781408436755024114785564396065520383254329807287031702570679\ 76588576905914420448144598194896011038291452214474271701118288728740364654817930\ 39268747655397553104665485315144777230865822547354781770364147960910195086807547\ 50607760070830458623876440801463579158285894505827276022017947676656024821321113\ 32045211860132786342979073276713224214531022696869286609999384788616280259521689\ 29044332295615957916635516143175107227411921906135979976139032498843271638305180\ 34742253157738296774385097451444482039037658289290322110834602457387576752868538\ 01628575425360702121619457584322245170300018948888989212101455690197146746427110\ 05415452040744879121015819232373232331630975941926945038777832005018304710257785\ 37472190071447978736873000222039901245302535469550065276529160819535952633490098\ 96145842080265041246742758836494394920830918701217240239167577530203183895348556\ 71708942489192020097269064492053373270355217798864551638040874667576710596443217\ 62650646567743522288753835279079839469914663102085562964522411825324475384570881\ 13012141726986439912563885452561384820988969801154192651050805220163646765300463\ 64557972997343108661574724631157670101221330584393137636815180974856270922739925\ 90357443698125194714724075059322102044036918915199289362856199936023459377757535\ 19482878256572278837844992924565391964546973596192329664800181239319065030693833\ 02379282053281456666409983630446861106682151212925357937965797138754350692550220\ 2556524675438321790153398980040564214165006920098729863564383 (mF+1)(2F^2+(r-m)F+1) is greater than n s is not zero r^2-8s=6783603622958388271272236871479188216731597472815806146823437768609904624\ 21074418746179071546702633210168861924308741343302193124641213150029425839914161\ 35799018038611952825454728120187717236797275431418919390967959767124145901206455\ 17012482195578184795271226524029671177792491833492622684366994867057133188265361\ 49678386054417144187570610144653083340326136092063615468800752603965320067352471\ 03120128761832132493429289505454569707518784679411221593157984921017199534149453\ 24262186896373182751479472636196140489221561169918903802187025057775056025893031\ 44482378916283235359000096027685295980084120025525175734793611986439841385548092\ 60991297819862621699198199915954921359006818998376051981689680383273417907819856\ 63907643534237451792061736694643182951335268597836160270150318677976602926598100\ 13583336185213434100099658716954769250129776647678332804648831233532021153529118\ 99441851989576175808043302034776710435357902355126063083584067684117192862643297\ 66828403454336904389647297895301778002093968943091931707739736262021716544492050\ 10758647630939540486862633277092734929957064187687550641676987169613060241050277\ 56281703515841803598599454294374002997544425086604320715955133946295822189830770\ 99869965902878626427116087643285678281540856519652836058076170650329927880014249\ 45365986997832439585013958952390000803595936948002200162045920370853718054769695\ 89040916187776292336415485853028204141357259003299159627767030586607153043562490\ 24185029515084512421392646302845947542556256565838888090669590497339295683903208\ 38082890911106700121385462056038973384194861908443271643018701540085124778347890\ 15706027072927200972128234326407337546922944316280204215390011540535866961184392\ 78746666774270660770242201031632778257912074911237702309286062491159060773374320\ 68944256997714338607624099502951770911284683629636299225877653413925570675550426\ 16489116291926478274466979258172632405142216275082631920639358198025179442826169\ 82442582195101308527771710485236616330066008895580414768846218002157156882141132\ 56102659154468371884442926711431007344618771296542986652743599730610089472348562\ 66932254383070414556434369488013412752085400166298721975091437663466312519426079\ 099105 r^2-8s=x^2+y, 0<=y<2x+1 x=260453520286410954689134425499126426107653040327387786774329330776753822793082\ 33164956300592782943964167487133545311455120145061044429991803172300718123360563\ 79361515899100672664895233026407831949834864988156380058927555292568297521515282\ 35802860507087303118338531540431042500999444733810381931164536459042696645045086\ 14036398099552835068342509696166141709274123170600776384047429943839011530904925\ 83883763760550493020558677203570141448905004784671616006582409077672648150935639\ 18467465323173862614231523163456871815573657522106811285200845344147158592157214\ 33056311907503426266153812393528048528714631784683181952754188143914655442161102\ 90000026828756610275074616316024795033922817296641205576327624083669989543351826\ 43349154130862122240819807578774608665493683689343438408113871455982193532755850\ 15504051572543223126233581058193748228239363097793112668553232107568581873648143\ 21639439260981089094627848161213336535453791461641301227212516006404263237855007\ 72331341638793773607439018077598420318242462082506049820678975772594714634336982\ 798314649477118995702257237062102401770956 y=520907040572821909378268850998252852215306080654775573548658661553507645586164\ 66329912601185565887928334974267090622910240290122088859983606344601436246721127\ 58723031798201345329790466052815663899669729976312760117855036610041658306806612\ 90005319768592483819781282250535838460572538635756157250192955726546814737635912\ 60936299355741322222888826303915895007716453159165281824992603287181301249139247\ 37877745862174070061210225292894291932952801953952939508588236942401408378725513\ 99272303170355726317119081626726771216881411517003863764133272058707913340783636\ 08345862605494235363074596934114889569188754240325096823235326600438934179278903\ 48251305303223114181459199609126618435463240178085025740705902286488988708960278\ 34564085918158263482292701894192946572152878529842041371328380087807428321757228\ 53279513923577245573383076729538194671367678070066875898856167458563409105096996\ 43274600712907171281644494787412224594610504105110608440880965558985785082597388\ 36584264177538031724258435824733955913695110915591966414974812105645750683501881\ 403183646590111670020343447280500993945169 r^2-8s is not a square n is definitely prime 108.516 sec.