Primality Certificate for (9926^2521-1)/9925

This certificate uses a theorem of Konyagin and Pomerance to prove an integer N prime by making use of a partial prime factorization of N-1.

Factorizing N-1

As N is a Generalized Repunit, we make use of the algebraic factorization of N-1 to arrive at the following 30.972010% factorization of N-1:

From this partial factorization, we use sufficient of the largest prime factors of N-1 so that their product F is at least N ^3/10 :

Note that all prime factors listed above have been proven. As primes of under 250 decimal digits can be verified in a few seconds, proof of their primality is not included here, in order to save space. Larger prime factors can take from hours to months to prove; certificates for all such factors have been PKZIPped into this file.

We set R = (N-1)/F. Note that GCD(F,R)=1 and Log(F)/Log(N) = 30.972010%

Finding a Witness to Primality

Next, we find an integer witness w such that for each prime factor p of N-1, w^(N-1) ≡ 1 mod N and GCD(w^(N-1)/p-1,N) = 1. In this case, w = 7 suffices.

Express N in base F

As F³ < N < F⁴ and N ≡ 1 (mod F), we can let N = c₃·F³ + c₂·F² + c₁·F + 1. Let c₄ = c₃·F+c₂.

• c₁= 896446975 4642040670 5316135621 8216698767 7581457546 0907724745 6930479295 5529268559 7552685116 5127658209 5827664512 6656222690 1930903713 4078133984 2731696712 8433682903 1086456877 3650012313 5054829789 1066548167 3673143081 3622706954 8555622905 0414255681 6485831329 5546853677 7025761237 1488644233 5481643685 4249780535 7275244165 9234603332 0954281341 0617783906 2141540733 8950829685 7559614926 6474367959 3783560635 9808854984 6982084027 7503828422 2885787669 9139361600 3176675395 6813235941 1730252530 4603127104 7375847318 5484952238 0806251957 5409333349 4375759724 2780007170 5444569077 2600412267 7647330369 5481367286 1224722393 9640871997 8534453446 1700608355 9596491148 9952470748 4469864847 7974679262 2972491201 4047298084 2626292807 1358188814 7232264125 7404848618 2280200693 9816662534 0695919249 2269270313 8207153048 9236518952 0578860743 0520400845 8354602906 8205675757 8004472910 7682554119 3545954466 0218810175 6448489082 9769399055 2996067054 2360577170 7030237519 5007266899 0654747908 0852991414 1228059497 4285287776 9762979484 2711556020 3855080640 5234080664 4741911115 5416938872 3155010295 4067230291 5166284371 9223001306 5459539845 6938642652 9328694467 3859250209 2693790835 6608138514 0367761693 6979008914 1848309212 6119210675 9097349755 1281684785 1474593000 9756397742 8193829280 0101812095 6397806190 3820010668 7058326422 6458523285 5324443991 6277021702 6555448528 3689386415 1668698434 1398454074 6166492091 6804816716 4876529536 1879688935 5839518389 3682354114 8992452361 6469934280 7909564752 4598493012 1653166405 7239146615 6450606717 1404194800 3886874583 6851311830 0985764552 8640305082 0618531555 6951886167 3388741848 7963818739 9193910745 8372592419 4053060883 5965176142 7667118269 7456783151 0820670127 6797375640 5539912567 4406382167 2004502335 8004479285 0589174484 5108956291 7787575970 8808641557 3921604059 5447313057 2590729113 1941838333 0524883820 1468916779 9730424265 1975096334 4274048201 3099723515 9083408355 0664568552 9090666489 3515497519 7843840378 2571074590 4735473470 7582325958 3547255195 0124591179 8534685870 1008614035 7392366637 0707936749 1779194359 2365669974 5587436343 6243617119 9242605813 5632803803 0749758202 9851063841 6503471956 2219362876 6176102149 7930852954 7050786562 1588745191 1745030230 4896837235 6898714464 6639827519 5604874737 4072963268 4137292578 5783588709 8404475899 5962349202 9807152032 1532675283 2055762548 7670031706 7420171372 7580891228 1673832549 5795775312 0557361345 6307821934 3118380758 3221128725 0138033849 9713679653 1902642873 2430331901 2423759853 9253672298 7407253973 6653075368 9796632599 2396073687 2164605478 1630063546 0857550266 0362579382 5215324033 0187942705 6806497487 0840362566 1937661056 6209281820 3542824990 4189080608 9819613939 9462436080 7549078210 8651153439 0195627000 2377039279 1944446565 3802915870 3732255809 3116862444 1866598239 8727289707 4282191627 4590846716 6525700211 7532269386 0414918474 3952954965 8201974544 0180764458 5659651398 9786751252 7439407407 0859010150 7821443809 4985808594 4807896164 6773645686 4516803319 8336033913 4878644584 5336554731 9831320200 3969267493 6355606585 5339877946 6541793013 0503383313 2790685746 7272531184 9813319313 7791799965 0610282417 0539199755 7639712961 8534489319 2247962348 7210923236 1187786819 5287540536 8393918344 4498343104 8284316153 5944897434 1222265887 7109252605 2216509956 4647986264 0645304217 9355898187
• c₂= 1256268569 3375033056 1121077094 6772059836 8457365642 2884850087 8468796169 7216251329 1167463167 3134304463 6168336106 0021870268 0409153210 1339046286 8123685177 1812597581 0483033079 7029721971 1481466738 7006560742 8376353307 6339567817 5184552219 7151099434 3528515803 8570101810 7050528607 2098045228 3827377354 9656673174 5439417119 4473957996 4391819011 6575827312 0856294933 6348243681 1606136096 2848635191 8000050023 3672539636 9528489888 5957359665 9879086880 4250284953 2172545097 9664638545 2933438175 9919443029 3141507288 5818769436 3440169939 9996268716 5031262508 0928813378 8601190155 8916993851 8977849620 1880730639 6817634060 1600150870 9300952565 7067962595 7044082148 8381247854 8537205766 0699767663 9028711940 8332394589 6258237361 8059043245 1922199880 3114207912 5468802682 2409472087 7740232492 3077430085 1428540631 6711066788 4283839910 2939804057 9618997712 1261275645 9875728364 6287960194 2219667327 8689994429 8833024627 2272312853 7819513131 5791509386 5723040547 7401230110 7375935990 6149647963 3175246514 4744672663 0664734036 9929017366 1363611505 1249998209 1094374407 5338987407 1282103313 4280492794 8327601308 1399650504 1127528579 3174384143 6072197777 2722838622 7049522787 7023360242 8926197430 3996920651 2037435744 1744169249 1714534577 9536445518 4657505568 4272396436 7385555823 2909831813 8728426086 2770160472 5643063799 5530052397 6550069665 6342253848 2148131520 6785992243 4065184129 6207732727 4846487705 4221722374 5634906150 5140840301 6149148491 5100817323 6217162826 1667763426 4117153959 1502906978 2935353486 6508103624 6968059898 9192323760 3582114645 3447345297 9470960818 3153214491 1877341907 1158450175 9402297684 2441385101 1327995118 8637697271 5821095476 6996405810 3252463809 9311142247 7625603568 7480582977 2747779101 4786324243 8832475195 9723969794 1104040519 6534806993 4772504967 1334485164 8192292543 8884322073 9710314258 7663407829 5711978817 6545771845 3846646710 6173446411 4705920981 1529015734 1061977171 6780282303 5191316860 3014105801 5665697776 4319643404 2578003771 4813425841 8875504171 3374219797 7378820976 1954100609 3437877040 9231469174 6391610644 7953781307 8853127404 4545777614 1773072147 2128649078 7833916779 2232182551 7871578761 9687735416 0774231518 6733611286 6237992848 7611045650 7679795582 8663261734 6634844548 9314632831 5153934494 6570927804 2083102025 5417087490 5193190962 3883452907 3894000098 8586989948 5819189888 5514498570 5020515610 0477893393 1957617858 7394830515 2964216170 3953016280 4660064515 8473458702 4439511524 5650126188 1022831447 2600385134 6046313066 9264918745 0452175617 7930141626 1059925881 1657538134 9531049768 3246688235 9325956294 6784868692 4123064620 2461018652 3261521566 4399954008 3643281823 6555270862 0813304259 1508405140 1593442176 4191540083 3718750917 7565478209 1113065996 2946681768 2933560353 8589826839 9069427327 6567745577 8655866906 0529736078 4333571411 1395493466 9636200704 7450222706 7563029481 7764188559 6614892607 0914289801 5482450238 9377452655 0180222551 6582812814 4532212085 2100114668 2153976673 2292586939 6979592905 2525429783 8441956940 2095983780 4453525732 9405082672 8698671206 2412873177 9477657626 1633204700 6884861780 9354761897 2965855751 8744011048 5315706774 2664090038 0644574541 8515647632 9265028417 9502136605 0897711424 6479163465 5643146255 2406490085 1420631951 5715189139 2894231410 2607806820 1667800011 6480020079 2226434550 9757672392
• c₃= 3077 5918657794 5700456618 2319063629 2181180500 6917757735 8022139477 5229576610 1001614988 4913028568 7582300518 9832938118 3283558774 9845172656 4831851162 6803349656 4463209060 3715847863 1664019590 3339966842 0636777956 7555204857 6434534720 8190766307 3584669639 8464107368 6891608454 1348009451 0856403075 5682037926 6226827941 2139807944 3410610229 6317802660 8432206221 7240957437 7020426813 3005200155 0887787871 0497066914 0274582126 5201754382 2849187732 1612779582 9006087125 2414326828 8914457137 0059190525 0221842112 9775270465 2180832551 1375042762 4138959374 2032658925 3184609470 1394269169 5716714301 7082387050 6155138941 9163832075 5524697323 4311828396 8743675621 5225316468 4749472617 4754840839 1074303633 5169318422 0282836260 1546611176 5748294781 2782788411
• c₄= 889 6300803353 2922159690 8921242426 0234673759 5760528623 5951954839 4595272523 1871133697 3463944735 5155073818 0882029119 7135164283 4279082768 1211981176 4357709367 7157745590 2783346379 0576409544 9411989030 1474826953 6748727952 4122503045 1600183493 8844105050 9866220961 8348037732 3718215558 3975337152 3836961611 2973805096 0423257390 3359950664 5054827320 8805831590 1493088357 3858822309 5185512658 3779805358 1384698629 2008664389 2641738530 4027561313 6228544045 4625032145 1068642746 0311190027 2401134653 5642330193 5666754680 7646299519 0263639568 7834690284 5074139357 9251352712 2408065852 0993747639 4340783759 4530750114 6024770271 2661059231 9061358828 9334004040 7659799147 5639503172 2026837449 5510493106 9069921863 5823679758 2454969947 9707881889 6762567586 7513808211 3182508000 0857761792 4926402733 1570634526 8283950713 7141365268 3094320918 4541803824 0080151897 8521497102 6108356477 2369031827 5643063762 6568549549 7717339465 7905389294 6346609164 1119518137 9875263717 0719866709 4871582120 4392597482 5812791236 2216277925 5385332680 9219395360 3380200238 1645221441 5270186202 0293629712 6547559831 4603215327 3261307429 3421759379 0342847675 0495241491 2713030515 0473055849 6264360599 2191878419 9847468591 1113488062 5670188433 0861223456 8372153323 5941454298 7680327308 6048571482 3519424303 5297160350 1705576184 3035883005 8846329897 7732264300 5264917300 1166472502 2284495473 1317287319 2774236998 1865371767 7871691611 2308852664 9599410741 0450745612 4946270816 7860515506 0941549656 1022931868 4881855991 7948212325 2718202789 8095655135 7313329427 9190544147 6207892045 6128660467 3286459588 0990958974 6377760345 2491543691 2466132771 7017218496 4484056158 3966931401 5033807162 6190917860 7441435864 4599211549 4757298302 4166899265 1259466410 0128008479 4890995427 4901331646 6715365272 7192370843 0783720645 8006394874 3815345693 1723640483 8308192471 5085340584 9844804935 1919502518 3079587875 2753849377 1533136838 9358210364 6973814240 2146922464 5500748490 7268813526 7947901620 3666520652 1105189959 5130158707 7244911204 9903834444 5490631713 4069359190 6529801377 9180387415 3017561790 5342974876 6110529951 5861927455 5856523572 2859850001 4026122461 5985071209 3284511106 3729271870 7036234219 5092837284 1260192981 0501546040 6596297373 7796563549 0381753515 7571461172 1801985275 1714710247 8886445253 3571007909 1961061134 6236706404 5676989328 1121843016 2936381790 7226617565 8468206486 4768285887 7301910626 1094330055 3102787757 8557091082 2717004936 6094176861 9036928971 1189239206 6319646089 7777436160 9306781139 4037135536 2353422458 4291156082 5266671965 8949343282 6612234336 5381013900 1623805537 6735851956 2956265658 6366571561 8916675262 3121487147 2539054128 0195114207 4610214199 7122636360 8200886698 9482310438 6717603811 7447548585 7751773158 6122452178 7394169518 2977237341 6017674366 1685711075 5319972957 1993701842 4013067348 2513450760 1799812466 1671240965 8697957515 4032631909 9220587808 2635287569 3228437962 8892469848 4955600066 4743685995 7446580098 4788425535 7756321289 6412818280 3487237405 1085629101 9093172136 0622767705 5444931171 2937323527 3167520044 3916654238 7121306824 5198178060 7715098670 4755287974 1976131943 3446771994 5031100675 8354377388 6869517070 3699300800 1696285632 1463690567 3483629621 8138007363 6270680609 2389482336 7958945251 6189748526 6271328078 3747561281 8404155346 0710227876 3934869108 8400846110 3666003292 4174214140 4060969771 6129481810 7758087190 9109765364 9526310333 2161642874 1255103222 1823022758 7320806965 1816527585 1688479560 6645876858 1438688626 0853932295 2629014616 2448116343 1574289027 6774556432 7291629773 8909345540 7565587103 4840177147 1335931557 4936845878 3161731921 1454438973 8482429437 5701126449 1066185160 9200081478 1129280354 5484046666 9491340823 3206154398 6400934161 6809568837 4004719154 6576222395 5156083943 4005585723 5834365879 2533689521 7215239176 4192949756 2388304006 3559176359 6079301774 8927973539 7911996416 4059083901 2038335155 8245292941 0645752243 5441137489 8996947709 4460352733 9077446709 3678776954 9712683450 9161684593 7045382227 1435306709 6413684302 0062377673 4725188187 9157025445 6454288619 1935900365 0592883602

Square Checks

For t = 0 to 5, we prove that Q(t) = (c₁+t·F)²+4·t-4·c₄ is not a perfect square. This is done by checking whether Q(t) is a quadratic residue modulo a variety of bases. If it happens to be a QR in all of the bases, we calculate s = floor(sqrt(Q(t))) and show that s² < Q(t).

• Q(0) is not a perfect square: it is ≡ 23 (mod 63)
• Q(1) is not a perfect square: it is ≡ 61 (mod 64)
• Q(2) is not a perfect square: it is ≡ 31 (mod 63)
• Q(3) is not a perfect square: it is ≡ 29 (mod 64)
• Q(4) is not a perfect square: it is ≡ 39 (mod 63)
• Q(5) is not a perfect square: it is ≡ 13 (mod 64)

Continued Fraction

We approximate c₁/F by a continued fraction u/v such that v is maximal while remaining less than F² / N^1/2 = 969 1560339749 6513724766 2342071877 0553042333 9160051009 3536820820 1356007672 7754071669 8989139068 7700073848 8054007439 2526005110 3089619044 7995707499 6537496274 2789905714 9807084650 5442003723 4937735651 6873316995 6998603548 0388567264 9254977266 9753543535 1054651022 9518351452 4483068024 0972117764 0281061331 8912212819 3631761184 5769017709 8153964133 7164431734 9716413486 2732920906 7653536323 5594662584 4254422016 7002237468 0391255043 4359117185 2546706421 5231624419 5273799822 2464159972 7875572363 8330145534 9266570250 3867968215 9017323151 6034394179 3545471818 7141380649 3649188741 3915250961 4193223634 9593675679 4135421228 8630301209 1346966138 0060411947 2517370425 5190065588 3058812505 4247362277 8903583555 0058581496 5425951699 8468632445 2405070004 7672284533 6334507653 7407050052 8287334555 7013009446 1625683230 7404619128 7520615445 8196657270 3278221214 6383742795 9114264971 5369699626 8623573181 0416292205 3202313075 1841507077 1140332960 8470050720 0419944361 4815228734 3353000439 8320612406 9405714972 4595441231 2341926327 3656220542 2716599229 3196149742 5771813487 3405982874 5311263804 5198470022 5851650961 5573592199 5662896142 9727768290 5226051124 8363149089 5626873303 7324372405 3423484370 2899778317 8932323123 5458859258 7795088565 6684183278 1575655997 7517806479.

With those constraints, the unique continued fraction is: {0, 3, 4, 2, 4, 1, 3, 1, 1, 3, 1, 2, 1, 1, 1, 3, 3, 22, 3, 6, 2, 10, 15, 49, 5, 1, 1, 1, 1, 1, 40, 1, 5, 1, 2, 1, 1, 11, 14, 10, 1, 3, 1, 1, 1, 9, 1, 1, 5, 2, 2, 2, 3, 1, 2, 3, 2, 2, 1, 9, 7, 6, 2, 3, 1, 1, 13, 7, 1, 2, 1, 2, 2, 2, 9, 1, 4, 1, 1, 6, 7, 1, 1, 2, 1, 1, 1, 1, 5, 5, 1, 1, 1, 8, 1, 2, 1, 5, 4, 2, 2, 1, 3, 1, 5, 2, 45, 2, 1, 44, 1, 35, 1, 5, 5, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 85, 1, 22, 1, 4, 5, 6, 1, 28, 4, 4, 1, 2, 2, 1, 1, 5, 1, 3, 1, 11, 38, 1, 2, 1, 2, 46, 3, 1, 11, 1, 2, 3, 15, 4, 370, 9, 1, 2, 10, 10, 3, 12, 2, 5, 4, 1, 2, 13, 2, 4, 1, 142, 1, 1, 1, 4, 6, 1, 46, 1, 3, 2, 1, 2, 1, 4, 1545, 20, 5, 28, 1, 8, 1, 1, 4, 4, 4, 1, 1, 379, 3, 1, 10, 4, 26, 3, 16, 4, 2, 1, 5, 1, 1, 5, 6, 1, 9, 1, 7, 8, 1, 11, 6, 1, 23, 8, 1, 3, 1, 1, 2, 59, 3, 1, 4, 3, 1, 1, 1, 3, 6, 1, 10, 10, 1, 1, 1, 15, 1, 2, 7, 1, 18, 2, 1, 1, 1, 2, 5, 8, 1, 7, 2, 1, 8, 2, 5, 6, 7, 1, 5, 3, 1, 4, 5, 1, 51, 1, 15, 1, 557, 1, 2, 1, 268, 1, 5, 1, 29, 8, 1, 15, 45, 3, 60, 1, 1, 2, 1, 1, 7, 1, 1, 7, 2, 3, 1, 1, 1, 5, 5, 6, 1, 4, 3, 1, 4, 1, 210, 2, 10, 2, 1, 3, 2, 5, 9, 1, 6, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 33, 1, 2, 2, 1, 1, 2, 1, 9, 1, 11, 1, 1, 17, 112, 2, 4, 106, 5, 1, 2, 6, 1, 6, 1, 7, 1, 23, 1, 11, 1, 1, 6, 1, 1, 26, 1, 15, 1, 2, 2, 7, 4, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 2, 4, 13, 1, 3, 2, 2, 1, 2, 1, 5, 1, 1, 6, 1, 1, 4, 1, 1, 225, 1, 6, 5, 1, 2, 3, 61, 1, 3, 3, 1, 17, 1, 85, 5, 1, 1, 1, 123, 6, 1, 23, 19, 2, 1, 1, 4, 1, 2, 1, 13, 2, 1, 2, 7, 1, 11, 3, 1, 198, 1, 5, 2, 1, 1, 1, 13, 23, 1, 14, 5, 6, 12, 2, 2, 2, 3, 2, 1, 23, 1, 181, 1, 14, 2, 15, 2, 23, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 4, 1, 16, 1, 1, 2, 1, 2, 1, 28, 8, 2, 1, 3, 1, 1, 3, 1, 5, 1, 3, 1, 39, 14, 7, 3, 5, 2, 7, 1, 6, 2, 49, 109, 8, 24, 1, 39, 1, 17, 5, 1, 2, 1, 2, 4, 9, 5, 1, 1, 1, 8, 33, 1, 8, 3, 1, 7, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 13, 1, 3, 5, 2, 2, 1, 2, 1, 14, 1, 6, 1, 1, 2, 10, 1, 5, 1, 6, 1, 50, 2, 2, 1, 1, 4, 8, 1, 32, 1, 5, 1, 37, 3, 22, 1, 1, 2, 4, 7, 1, 2, 32, 1, 4, 2, 1, 1, 6, 5, 2, 1, 8, 2, 6, 20, 2, 2, 6, 13, 14, 2, 1, 1, 6, 1, 76, 1, 1, 2, 8, 3, 1, 2, 1, 1, 9, 1, 4, 2, 1, 1, 7, 1, 10, 13, 1, 1, 1, 6, 1, 9, 1, 1, 1, 11, 1, 5, 4, 5, 4, 1, 5, 3, 1, 1, 3, 130, 2, 1, 1, 3, 5, 19, 1, 2, 14, 1, 2, 2, 3, 1, 2, 1, 32, 2, 2, 8, 1, 2, 6, 2, 1, 1, 1, 3, 169, 1, 1, 27, 1, 15, 1, 308, 1, 9, 15, 1, 5, 87, 4, 2, 6, 4, 2, 3, 1, 2, 1, 30, 1, 15, 2, 1, 8, 1, 1, 2, 1, 3, 1, 1, 6, 2, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 6, 1, 1, 37, 1, 14, 2, 2, 2, 8, 1, 4, 3, 8, 1, 2, 1, 6, 1, 15, 94, 1, 1, 2, 15, 1, 2, 1, 22, 2, 1, 1, 1, 7, 2, 1, 1, 1, 1, 1, 10, 2, 2, 11, 3, 2, 7, 38, 5, 11, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 12, 1, 2, 2, 25, 1, 5, 2, 1, 1, 6, 2, 1, 4, 2, 1, 6, 1, 1, 5, 1, 6, 2, 1, 33, 4, 2, 1, 2, 1, 3, 3, 1, 2, 4, 1, 2, 1, 2, 12, 1, 1, 5, 1, 2, 3, 2, 16, 1, 1, 2, 3, 2, 1, 2, 1, 2, 2, 5, 7, 21, 26, 1, 4, 1, 2, 16, 1, 1, 6, 1, 10, 2, 2, 1, 1, 9, 1, 2, 3, 3, 5, 5, 4, 15, 1, 2, 8, 5, 8, 1, 4, 1, 10, 37, 11, 2, 10, 1, 1, 4, 1, 1, 6, 2, 1, 1, 14, 1, 24, 4, 1, 1, 5, 4, 1, 19, 34, 3, 2, 4, 5, 1, 1, 1, 2, 1, 1, 42, 1, 422, 3, 5, 2, 3, 1, 1, 1, 10, 1, 1, 1, 4, 12, 1, 3, 7, 1, 2, 1, 2, 3, 2, 4, 3, 2, 21, 23, 22, 4, 1, 3587, 2, 1, 1, 1, 1, 1, 6, 8, 3, 7, 6, 2, 6, 1, 6, 1, 3, 1, 1, 1, 75, 5, 1, 11, 1, 1, 5, 1, 16, 4, 2, 1, 7, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 4, 2, 18, 2, 3, 4, 2, 2, 1, 7, 4, 2, 1, 8, 1, 3, 1, 3, 7, 2, 2, 3, 1, 2, 3, 7, 1, 30, 1, 2, 1, 14, 5, 3, 4, 4, 1, 1, 1, 1, 1, 1, 4, 1, 16, 1, 1, 7, 1, 1, 165, 5, 339, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 5, 1, 2, 4, 9, 1, 2, 1, 4, 4, 11, 7, 3, 2, 5, 1, 292, 1, 4, 1, 1, 3, 2, 13, 1, 5, 2, 1, 34, 2, 1, 5, 14, 8, 4, 1, 2, 11, 1, 1, 12, 1, 1, 1, 1, 2, 5, 2, 6, 1, 8, 1, 9, 1, 2, 2, 7, 1, 1, 5, 18, 3, 2, 1, 4, 8, 3, 1, 1, 1, 3, 3, 3, 1, 3, 2, 8, 1, 3, 8, 2, 2, 1, 3, 1, 1, 2, 4, 1, 32, 1, 9, 1, 8, 1, 1, 1, 3, 7, 23, 2, 1, 3, 1, 6, 18, 9, 1, 33, 3, 1, 2, 1, 2, 9, 5, 1, 1, 1, 5, 1, 2, 1, 2, 1, 10, 1, 1, 10, 3, 1, 3, 1, 4, 1, 1, 1, 2, 4, 1, 2, 1, 12, 6, 1, 1, 1, 1, 1, 2, 4, 3, 3, 2, 1, 20, 24, 2, 2, 2, 2, 7, 7, 1, 210, 2, 2, 2, 2, 1, 15, 1, 9, 11, 1, 10, 2, 12, 1, 4, 1, 3, 11, 5, 7, 1, 2, 4, 1, 1, 5, 3, 1, 2, 1, 13, 2, 1, 3, 6, 3, 14, 1, 4, 3, 1, 1, 11, 1, 4, 1, 1, 15, 1, 1, 1, 4, 1, 2, 7, 4, 1, 4, 2, 9, 18, 38, 3, 1, 1, 2, 2, 1, 16, 1, 2473, 1, 3, 1, 2, 2, 5, 2, 1, 1, 4, 8, 2, 1, 14, 1, 2, 8, 13, 1, 2, 1, 4, 1, 1, 2, 8, 2, 17, 1, 1, 1, 4, 1, 2, 1, 31, 1, 47, 8, 7, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 39, 1, 27, 15, 6, 1, 1, 2, 3, 1, 1, 2, 1, 3, 2, 10, 10, 2, 1, 1, 2, 1, 21, 8, 2, 1, 1, 1, 2, 2, 2, 2, 1, 4, 1001, 10, 1, 11, 57, 2, 5, 1, 40, 1, 1, 6, 1, 1, 1, 2, 5, 4, 4, 11, 3, 1, 11, 1, 3, 3, 1, 2, 8, 1, 10, 1, 4, 2, 44, 2, 1, 6, 35, 1, 9, 1, 2, 4, 1, 1, 1, 2, 2, 14, 2, 1, 11, 1, 2, 1, 3, 55, 1, 1, 3, 21, 3, 2, 2, 1, 4, 38, 6, 2, 1, 15, 1, 3, 183, 1, 2, 3, 1, 17, 1, 1, 7, 6, 1, 48, 5, 3, 1, 11, 2, 6, 1, 3, 1, 7, 41, 20, 22, 9, 17, 16, 2, 7, 17, 1, 2, 19, 2, 1, 3, 1, 62, 1, 2, 4, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 4, 5, 1, 9, 1, 29, 21, 1, 1, 45, 1, 12, 1, 1, 3, 4, 25, 1, 2, 1, 1, 2, 4, 1, 11, 1, 3, 1, 2, 1, 1, 6, 1, 9, 1, 7, 2, 4, 10, 1, 1, 1, 1, 3, 3, 1, 45, 1, 10, 2, 1, 4, 1, 1, 2, 1, 4, 4, 3, 1, 2, 3, 1, 1, 6, 1, 16, 1, 1, 4, 2, 8, 8, 1, 1, 8, 1, 1, 1, 2, 1, 1, 2, 4, 3, 1, 1, 1, 4, 1, 1, 1, 81, 2, 1, 1, 2, 1, 2, 6, 1, 1, 1, 4, 10, 1, 2, 1, 2, 1, 1, 23, 2, 1, 1, 1, 1, 6, 1, 2, 2, 3, 1, 2, 2, 1, 5, 3, 5, 1, 1, 1, 1, 2, 3, 1, 2, 1, 28, 1, 14, 1, 5, 1, 9, 8, 2, 1, 4, 6, 1, 1, 1, 3, 2, 3, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 3, 2, 22, 64, 1, 1, 5, 5, 1, 118, 24, 1, 3, 1, 22, 1, 1, 6, 1, 2, 17, 3, 7, 1, 6, 1, 1, 1, 1, 1, 9, 1, 1, 11, 1, 3, 1, 2, 7, 770, 1, 6, 3, 4, 8, 6, 2, 1, 11, 32, 1, 1, 123, 3, 1, 5, 1, 1, 5, 1, 1, 36, 1, 638, 2, 1, 36, 3, 9, 1, 5, 17, 1, 2, 3, 28, 6, 1, 1, 5, 1, 1, 1, 3, 1, 11, 3, 3, 1, 1, 2, 2, 5, 1, 6, 8, 2, 1, 2, 1, 1, 2, 1, 33, 1, 4, 1, 1, 3, 5, 20, 2, 2, 2, 4, 3, 2, 3, 2, 1, 3, 2, 1, 3, 2, 3, 1, 2, 66, 1, 7, 10, 3, 4, 11, 17, 7, 1, 7, 1, 2, 1, 8, 1, 44, 1, 36, 2, 1, 2, 2, 1, 3, 1, 2, 7, 4, 1, 1, 2, 4, 1, 10, 2, 1, 1, 1, 378, 2, 6, 15, 1, 2, 27, 1, 2, 1, 1, 2, 9, 3, 1, 9, 25, 1, 1, 2, 11, 4, 1, 3, 5, 3, 6, 1, 3, 2, 3, 1, 1, 8, 4, 4, 1, 4, 1, 2, 1, 2, 1, 5, 2, 3, 2, 12, 1, 51, 1, 2, 1, 1949, 5, 1, 1, 2, 10, 2, 15, 7, 3, 1, 2, 1, 1, 1, 5, 1, 3, 2, 2, 1, 1, 15, 481, 1, 2, 25, 1, 2, 27, 1, 3, 2, 1, 7, 2, 9, 4, 1, 1, 4, 1, 3, 1, 1, 2, 1, 5, 4, 7, 1, 1, 28, 1, 3, 1, 3, 1, 1, 1, 1, 5, 2, 1, 8, 2, 1, 1, 1, 1, 4, 2, 7, 3, 1, 2, 14, 1, 28, 1, 5, 1, 2, 4, 3, 2, 2, 1, 1, 1, 1, 1, 6, 218, 4, 1, 3, 1, 1, 2, 1, 3, 1, 19, 1, 12, 1, 2, 1, 3, 6, 7, 5, 2, 8, 3, 2, 2, 3, 7, 1, 10, 1, 3, 2, 4, 3, 1, 1, 8, 3, 1, 5, 15, 1, 1, 1, 3, 13, 1, 2, 6, 27, 7, 1, 1, 1, 4, 4, 2, 1, 2, 8, 1, 2, 1, 5, 1, 1, 14, 3, 25, 1, 2, 3, 5, 2, 3, 8, 14, 2, 1, 3, 8, 2, 1, 11, 5, 3, 3, 21, 1, 6, 79, 1, 1, 1, 1, 5, 1, 2, 14, 1, 2, 1, 4, 6, 5, 6, 1, 4, 1, 1, 7, 1, 10, 11, 1, 4, 1, 1, 4, 11, 2, 7, 3, 1, 1, 3, 22, 1, 89, 3, 1, 1, 2, 3, 3, 9, 8, 1, 40, 5, 2, 93, 1, 13, 1, 1, 2, 1, 21, 8, 1, 213, 1, 2, 1, 2, 1, 1, 2, 11, 3, 10, 1, 2, 3, 1, 4, 1, 1, 1, 9, 2, 2, 2, 2, 1, 5, 1, 2, 1, 1, 1, 1, 11, 1, 16, 1, 3, 1, 3, 9, 3, 3, 1, 1, 6, 3, 13, 5, 1, 2, 1, 1, 7, 24, 1, 1, 2, 10, 1, 1, 1, 1, 1, 14, 3, 2, 2, 18, 1, 5, 18, 1, 4, 1, 1, 7, 6, 1, 5, 5, 9, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 11, 4, 1, 8, 2, 22, 1, 2, 2, 51, 1, 2, 1, 1, 1, 3, 3, 3, 2}, giving these values for u and v:

• u= 188 3328673251 3587800968 4769281549 2425361009 5095155684 1062244344 7169252989 9464934504 0710003967 0665679873 7897424594 0338513168 2185985370 2503947644 7614633916 7807115190 0372803206 0593108022 4382331901 8693826779 1193946575 8650854059 0604677429 0588217573 7473201015 4800732921 7965044698 4530132323 9307915772 1919841902 5173271409 7526902972 4735746926 6818545544 3429129381 3603491285 5233189546 0913160399 9270565170 4908454601 3532452371 7505908475 3091900142 9649420753 5047668964 7470226491 0954864953 4896821277 9137566635 6540315229 1550731275 3266255011 2583340800 8611406296 5260570726 5389311048 9101531462 3670665229 7083839081 1234754472 5054647757 1890912767 3446816677 4736514590 2638546257 6041201037 9556147782 7764362373 4496275853 5865536057 1445215049 4106480036 6674724610 6125029947 5560613647 9018096488 3147151175 5299432632 7504412204 1390985489 4141202197 9786433764 0310251743 5898780075 4230326162 3881150800 8849683531 7040110030 3598341387 3102808794 3069550039 0771922459 6831052012 0082912127 7779512392 8278513213 5787008042 7126962298 3306065437 0026397471 9144508073 7229638448 2629831404 3700473111 6866845884 0211406342 4524685047 4532824931 3521724229 4618829275 3441066508 4569624556 6445846070 2799635253 4973020463 5717739088 1545936802 7310380188 0352152218 4791125630
• v= 607 2953358887 6670162169 4283021292 4294838518 3329117147 0252843307 6532565696 7412649323 5722485244 7753506753 6007010555 4635055036 9015960512 4576656443 7342597772 8754338662 9000218076 8379280905 1651924693 5043589948 9626267410 6391091794 8044300771 4069410421 4258335683 5736722178 9299393395 8278041914 3425892277 1448518426 7372994108 2728908064 1354484653 7219594670 4273261034 4045586819 3613680747 8520795387 0320339842 2559666206 0059052350 7744857163 1754845881 4823872672 5794424146 4822926415 9765122981 7586206733 2814945036 8977138795 4142386112 0784586668 2462201296 1169400460 8882126064 9203164127 2425342777 1799802075 9056243590 3006066796 0756571337 1349553546 6575307617 3930958954 8763741916 7584278832 5877123230 7071793802 1698972332 8473179994 3131323642 6423272096 7576907413 4638880537 4037721927 5130154800 0723311462 0838629987 9249979125 0758986108 4296957460 7139389405 2863234251 0975131048 3652966102 5687619583 0914890647 9316287315 5025713925 9292958941 1936808417 9893274235 2435730715 7154634402 6906860332 8928737700 7222422124 7414462567 8969487284 1931456095 0022707348 9565330154 8781075906 1984366030 0946590354 5935004843 8931921058 8518384581 8461135366 1242766869 9605711535 6378001671 6915213840 3242463709 9153839553 3266694156 5934937065 4074381959 5430277700 5539738717

… as taking one more term in the continued fraction would give a value of v of 1478 3407908951 5906263107 1047181103 5759350740 4984322617 0084524972 1533265782 8833789491 0605106762 6031394336 8433596813 1275598979 7006068583 3936134516 9363763323 5170699851 2861420715 4774609196 7290070029 3849437398 5205027050 5070118381 7710272382 3577195191 3674311011 3513644643 0957686521 8948051102 6290798031 2876072287 8322333226 8645141314 6263143673 0355524350 7564411528 9002431117 0034617165 0740338988 2740895966 3617175436 2552256998 3096462411 1380752633 7228484723 2070185329 7961115714 6982400869 1144417269 6799131503 2352458410 4153697604 2580956868 8472430498 1688152642 7101782548 4484567876 3616885022 3247387158 3631797090 0802629065 3625967977 4011655706 0519474780 6182708508 5056597095 8374042799 2369274696 1922441115 2681720348 4271593848 6615022397 8538818495 7183470056 9884915742 3109567529 1318495796 5033258966 6779115292 1325060986 9397654389 6612833991 1275018319 0006745351 8784234458 6134861946 7097580815 7283208477 5831974945 3410141178 4277407416 9723966093 0991285933 2556455257 9989857774 9085463038 3049206675 4759856705 4040383862 5312316536 7189495490 6546805069 2544154521 5463055306 3385053376 8365957722 9470325411 1512427683 2947085519 4426346445 5277267538 6696685429 1849361132 7237148271 4901626515 4152248139 8208459311 9408117460 5856124363 1511215480 8966838031, which is too large.

We also need to calculate d = floor(c₄·v/F + 0.5) = 1869007 1858570715 4919268028 5074372554 5767859627 8716430943 0949145468 1157870663 0560595395 9909542930 3110658580 2598601171 6738042189 5993199919 8649167893 5982245271 4936247568 9485632218 7204238698 8669063959 7185902202 7476164921 3361266778 0661429826 7536199867 7858048463 7359040971 9317699132 1034007070 7771999057 2507802701 4625077977 9450395143 4425078377 7945874942 1525000596 3460727490 3230145204 4893914490 8303690769 9634171447 5801951449 1684114519 3736500559 5596385222 0441996273 2530338011 0737886541 1848415223 8842059334 5555614928 7816986365 8518190395 7066002175 2424068860 6236382690 2697486497 2903058075 2561762311 3891028408 1492853207 4515100759 1054173725 0485037876 3521550874 0183304289 0291815606 7728502787 1342190642 4067390117 0946689574 5571914505 8250132903 2648909730 7652761909 8146571044 3025410332 0176351631 1681164743 4586045422 2724690875 7354697264 3562360728 2884593048 7570716769 9724885654 7401539344 1194374287 8801678716 3648554518 1684488074 5737489114 2139538480 6549542893 8880991664 6300011985 2729561105 0879682687 6840910374 3410642783 3994786096 1066989857 5860112628 1466438324 1910420769 8063045667 2656385163 3973232084 2551160369 4077918545 1116085637 6370259924 7329822529 0923085893 1838101323 4032675024 3134666579 7343386857 7342748764 4820816742 1283567163 4885967056 4229648022 5517132164 9430169215 1890959550 3584104920 1783805722 1825742201 2292859992 6248317319 5486393607 4203338124 3581523061 8564680097 2704270958 6637355206 4715962292 4115694500 6212274023 3380180779 9498976893 8012224571 5732719215 1828376213 6880424531 8006623021 2740021750 7920287536 8735818206 4829875272 8982569004 3976547052 7501435156 5583461178 1906963650 0304588046 1689123199 4720147074 6749391321 7976445844 5596246316 6950979782 3630988009 9364077937 3993961106 6772900046 5081882141 3366785110 3015669228 6945644499 9177375152 8754574011 0520070036 0845309088 2976674461 9480074308 2472633379 8109663160 3971666354 4448439522 2887093700 7484819049 8131371955 5786369536 0982517432 9838342403 7982345651 2993033363 8825835066 6931531743 7090373801

Cubic Polynomial

We now consider the cubic P(x)= v·x³ + (u·F-c₁·v)·x² + (c₄·v-d·F+u)·x - d, which we express as: z₁·x³ + z₂·x² + z₃·x + z₄, where:

• z₁= +607 2953358887 6670162169 4283021292 4294838518 3329117147 0252843307 6532565696 7412649323 5722485244 7753506753 6007010555 4635055036 9015960512 4576656443 7342597772 8754338662 9000218076 8379280905 1651924693 5043589948 9626267410 6391091794 8044300771 4069410421 4258335683 5736722178 9299393395 8278041914 3425892277 1448518426 7372994108 2728908064 1354484653 7219594670 4273261034 4045586819 3613680747 8520795387 0320339842 2559666206 0059052350 7744857163 1754845881 4823872672 5794424146 4822926415 9765122981 7586206733 2814945036 8977138795 4142386112 0784586668 2462201296 1169400460 8882126064 9203164127 2425342777 1799802075 9056243590 3006066796 0756571337 1349553546 6575307617 3930958954 8763741916 7584278832 5877123230 7071793802 1698972332 8473179994 3131323642 6423272096 7576907413 4638880537 4037721927 5130154800 0723311462 0838629987 9249979125 0758986108 4296957460 7139389405 2863234251 0975131048 3652966102 5687619583 0914890647 9316287315 5025713925 9292958941 1936808417 9893274235 2435730715 7154634402 6906860332 8928737700 7222422124 7414462567 8969487284 1931456095 0022707348 9565330154 8781075906 1984366030 0946590354 5935004843 8931921058 8518384581 8461135366 1242766869 9605711535 6378001671 6915213840 3242463709 9153839553 3266694156 5934937065 4074381959 5430277700 5539738717
• z₂= -1853486 6733782001 9847419757 2406606332 8363941256 1207271573 8273098656 0405395667 0625263797 9855494226 5665793653 3929781415 0780676845 7827732305 6364280163 3197806088 4908931010 6403988777 3217812505 7257344563 9967714004 8471842978 1677580456 7288959204 9763891867 3206101695 9100090377 2439170902 7125493631 6481304060 6514624945 3026639392 8322772273 9260474953 3053138562 5032498683 0842828938 0790308852 2185509693 0233566895 2878718951 1979100575 6383468115 2270930562 6150628765 4932708278 4646804372 4263817657 3508299800 3402688748 5584843319 7343626160 5368957222 6995787439 9228805499 6333973320 5964998049 2999877668 3881845014 4930054637 8948569779 7179461751 9451655126 9662059934 5945847604 9433198710 6695714467 9990620073 2105427477 8048484621 6872179251 4062636304 7325627509 9348243890 7082979605 1428845065 0129722636 5533635015 0585694470 3746703916 1919002347 9867427536 6273518581 8136128036 8082797482 8113217545 4613158469 3830147945 0750055537 3296053282 2552759950 7592080040 5847025111 7424001160 4767401036 5923025382 6038576296 0292572858 3235244549 9352786405 9460215545 1897213774 0099985184 0471135566 3450793984 0592286381 8016116727 5181882340 0316934260 6316351523 8691996017 5039186859 4029854382 1222053020 1763758538 5555935547 2012732428 8073934023 0010058409 1491743968 6017527223 9053294472 8680503746 6810036543 5632977073 7547208469 7047566336 2695986897 4608973562 3800484442 5066579893 7506059661 9266649607 3646493533 0879745107 9717792169 1338700880 9073775479 5518558685 1108335772 6170366364 4365280758 1822086182 4825918760 4834874476 2186807942 3012171852 9331110635 6801702099 3869919073 2692436764 9869164889 1109443345 2263760137 3445066298 7540112276 7223487938 8745456495 3184986097 6396748799 9893582856 1568419168 2264527718 4828551330 6083540490 6443926416 0753042352 8386698977 9357050401 9832399782 6428124699 8082567363 2749538278 8705990234 6151933836 1799113445 1692386113 4219551529 9091350202 5859709523 3006404779 4039251943 5886280144 1058061357 7330609790 4021390294 7164005934 2378083451 4771266002 2182185907 2997169706 6709806779
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• z₄= -1869007 1858570715 4919268028 5074372554 5767859627 8716430943 0949145468 1157870663 0560595395 9909542930 3110658580 2598601171 6738042189 5993199919 8649167893 5982245271 4936247568 9485632218 7204238698 8669063959 7185902202 7476164921 3361266778 0661429826 7536199867 7858048463 7359040971 9317699132 1034007070 7771999057 2507802701 4625077977 9450395143 4425078377 7945874942 1525000596 3460727490 3230145204 4893914490 8303690769 9634171447 5801951449 1684114519 3736500559 5596385222 0441996273 2530338011 0737886541 1848415223 8842059334 5555614928 7816986365 8518190395 7066002175 2424068860 6236382690 2697486497 2903058075 2561762311 3891028408 1492853207 4515100759 1054173725 0485037876 3521550874 0183304289 0291815606 7728502787 1342190642 4067390117 0946689574 5571914505 8250132903 2648909730 7652761909 8146571044 3025410332 0176351631 1681164743 4586045422 2724690875 7354697264 3562360728 2884593048 7570716769 9724885654 7401539344 1194374287 8801678716 3648554518 1684488074 5737489114 2139538480 6549542893 8880991664 6300011985 2729561105 0879682687 6840910374 3410642783 3994786096 1066989857 5860112628 1466438324 1910420769 8063045667 2656385163 3973232084 2551160369 4077918545 1116085637 6370259924 7329822529 0923085893 1838101323 4032675024 3134666579 7343386857 7342748764 4820816742 1283567163 4885967056 4229648022 5517132164 9430169215 1890959550 3584104920 1783805722 1825742201 2292859992 6248317319 5486393607 4203338124 3581523061 8564680097 2704270958 6637355206 4715962292 4115694500 6212274023 3380180779 9498976893 8012224571 5732719215 1828376213 6880424531 8006623021 2740021750 7920287536 8735818206 4829875272 8982569004 3976547052 7501435156 5583461178 1906963650 0304588046 1689123199 4720147074 6749391321 7976445844 5596246316 6950979782 3630988009 9364077937 3993961106 6772900046 5081882141 3366785110 3015669228 6945644499 9177375152 8754574011 0520070036 0845309088 2976674461 9480074308 2472633379 8109663160 3971666354 4448439522 2887093700 7484819049 8131371955 5786369536 0982517432 9838342403 7982345651 2993033363 8825835066 6931531743 7090373801

We need to prove that this cubic has no integer roots r such that r·F+1 is a non-trivial factor of N. Clearly r (if it exists) must lie between 1 and R.

P has a single real root at:

• 0+ε∈(0,1)
  (Root is not an integer)

There are no integer roots of P in the interval (1,R), so the proof of primality is complete.