Primality Certificate for (2409^2521-1)/2408

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 32.295995% 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) = 32.295995%

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 = 17 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= 6 7674846123 3254855257 6522281553 0228011172 7251809241 6911121789 0969606893 6125145818 8266084747 1442972489 9729697302 0539989645 0824430175 4932426792 3503016597 3740458745 9082250968 9772966388 3112152170 7726565918 2398833103 4822361030 0473944240 9564421639 2411688660 8445374491 6172230708 0260358052 4877023531 0951126013 1096515019 4580065301 7556539445 2610715139 6255297514 9401936624 0980397668 8674579908 8604410144 2382850473 2284273651 0604334243 5398289197 1992422436 9177187259 4944397687 3644947881 2300050475 9036717005 0164706077 3212110245 9583292227 6672040696 0338679251 7677331290 4256215013 1898587402 3233779494 1856427408 4335821930 1466620292 0843377699 8838393756 3455882451 7686729931 9814121220 5922562820 1479720958 5411053237 2475961052 6939465916 8610820937 6336064759 3702367909 4767689012 9810277987 6221541097 9476479927 9219272749 7076875589 1826168101 8871720864 7618280511 9996554011 3837223554 4151097977 8541285365 9715711699 2202654731 9439371784 7154877162 2953823819 8689408856 1983628474 7499614206 0306822563 6652881104 4108603388 7370327149 3999174519 9175520945 5439283171 3027040277 7808916077 9063244671 9046939432 4183447487 8094701346 2832685837 1094532117 7452738045 6635882498 1813033910 6068080311 8952811962 7964830094 2966942136 9787796808 0669185937 7288954186 0945233494 4654875340 4973093890 1391471762 8462312859 5922905173 1198045310 1946452250 8210530901 3123456625 4932775627 9785892727 5703369246 8684115936 2458810470 6371157582 1919184020 4868091464 5297327997 5000711348 3496798389 8760299972 0844615194 2377943639 6846577036 8282190460 0641051504 9140787694 0270297012 2794503124 7005823939 4467188187 1635270855 4196309301 3968991756 5304947921 1966491259 9501399547 8954046532 6713387630 7759545039 9881461978 7789389068 2519375388 2231756808 0889570421 6108830301 1608621340 7094455989 3488727578 8407021313 2509028566 2159900274 9788417970 3528099009 1791927601 3183234397 1339960177 0868135089 1514035873 2462678524 4510127350 6637774757 2257238393 3654806643 9932382377 8614424462 9375412852 8183783809 4718507460 3061654823 0802194279 7024745053 4190738310 0035233666 6042685840 4182464631 0214746401 6968598981 4215739617 2496478336 1820398662 7343189279 2743920413 4042630184 8873893496 7923737024 2464284918 7402053754 8828017917 6176592875 5803735286 2875094143 8609456280 5572615255 2653021881 4928457848 4551130197 5660408625 0167526097 7025810446 5243881536 3234919053 8096726814 2684570638 0674692092 4855876003 8751955243 4349099335 3810218059 0036758484 4799093263 9053979930 3352544632 3494718182 0617018894 3369444797 6958949704 6347949075 2181547092 8912283690 7045042776 7621539223 5319974482 8863823630 0107320259 5117284675 7525588354 7408615067 5740276587 8874251352 2714178048 5271444819 8782768555 9985714398 7514598939 9561147838 9486115548 2838709189 4113803527 1171670510 9945440317 3188116782 1535926315 6511136740 4390703185 9219269092 3702310205 9794955747 8136498131 9335323535 3628474988 5386763596 4114422019 5986777813
• c₂= 118 7672094766 0702181948 3597146698 1506790902 0272055210 0667207328 4778955989 0447789463 6544123858 5295248411 9278709599 4334677440 2963882042 1684177894 5658458080 4317292725 6433270761 0280916323 3385004916 7015605853 1380740565 4346684053 5138267890 3000869970 3322836385 1482624255 6953883204 3016990917 2262685949 7094890888 2349489955 6362212576 0801234078 2823252990 8375200617 4995867033 0394495764 9134647313 6940166054 5278154029 7513367483 4936272737 3047237665 1560763974 7217375976 6272031811 2655326103 7748895662 0105126126 7586040924 4069304156 8561807700 1252778959 6730109977 2105449670 3407339258 3690684332 9543974170 1110338867 5095309712 5221351183 4993908940 4430920292 1175129564 2801583011 8007241481 6987439435 8713593345 4425632145 0873698622 3314458342 5237827991 5496070157 6416115235 1098965988 6632211083 6935702112 0354597218 9459269183 4582374961 4832422561 1163020430 4968884109 0340510748 9534177419 9841697445 0753805147 2094964729 7194239993 7746172439 2752379809 6272396735 0901472393 3550240248 0846298537 5936305543 4035310474 2036177203 5625831080 5507275129 1345734671 2689268548 5238274662 5033994761 3650196697 7755720443 0594965609 1291277612 2120745573 1973850750 0291917585 9326636337 3465539236 1419191441 6372619561 2011623974 0890437172 3581777517 9201305715 7602994459 3455607173 0950658818 2124634833 5647184115 8285020856 7178127371 4503067711 7428163794 2447385937 2341352535 0857917712 8347972507 0200133482 0490601297 9794540333 3097256911 7333625419 0064601203 8785906562 7588786127 9259749816 2303992430 8609587527 9115989467 6824691771 7188169087 3296474751 6137809565 5526735195 3198340950 7888106203 4450149498 0187443282 9516895780 7864815976 2549502610 1764840672 0756468906 6943964774 1399606525 7979498456 4787573737 2221792716 8114068241 6515352640 4104325077 0102791360 6598028222 8670023261 9028698236 8947160115 9313461539 4162571636 2813542818 4147179665 0959894550 2470813205 5288094813 9121098978 6674854757 2181041175 8767932609 8471413321 7481720518 5063947996 0595288758 1160775068 7197798479 7387309043 9977046794 8474376455 1109541593 6986976982 4939020757 1147096637 4462604772 3842850729 2723574946 7614329415 6644474231 0093084031 2947339711 5036048014 4453161172 0257674814 2800278811 4949460109 3176425826 1323729043 0131452459 0572699668 7444498226 6649683186 9252666635 6271686466 1910161802 1854778121 5146871096 7523002379 0261915828 1654062637 8442887451 7109060669 5685898234 9827941762 2943347421 0913358395 8781654229 0387232921 7199486412 9870268311 7981481808 1370379387 7797986028 0582222989 0097681096 0479512673 7567271830 6222738728 4550881575 4227330149 9167329985 8769368718 4800670244 6566842491 1156911747 2625130188 2889396951 3666741126 2234608547 0293361644 5372618706 4469454052 3185477312 5348802296 2731580691 3921047612 7827070735 8052193801 8181623610 1595408133 7727790879 0964575193 5929133146 7375913045 1354596243 3023692461 5749783938 5030154554 0433521958 9696654238 4383533253 0802192759 4682082294 8717793342
• c₃= 163308 7582410787 2932227225 1575755481 7359875377 6042648308 7423186487 1866291315 5050615158 5598334608 4890065541 0225542308 7162349933 6497138265 7672054087 6357829468 2873858808 9888068585 3699189798 6114311615 3511535054 7037271498 6608125407 9065978352 5853080750 9478501535 0512312104
• c₄= 35615995 5819660579 6793025399 3287068451 4608705063 3151131205 1191003387 6410225239 5577248334 6643351336 1849732704 9429476645 9006428564 1271609182 3956196853 4887587127 1321522972 9146580415 2944877654 1717763502 6633017401 9996718022 3780224854 6073445705 2963643561 4587524981 2656050393 2525179932 1978947885 6696613737 9820840399 0520899772 5623428340 3984586662 0961639402 6320609839 2466378933 0879768889 8769633950 6806182864 1699352470 7092913702 3557599828 5649836635 5446615981 2462252496 6693678444 2663503921 0147918526 0148491916 7390767710 0752742274 5366064363 6963177003 1766028202 8499947445 7323525416 3536204613 2236497030 2762648036 5183002929 5150219944 1517865205 8960607520 8685738848 9557200270 6103394676 5075368383 5774682817 0599989236 5301403424 4022212726 2138184198 1060891749 3433064425 5272798730 7031744156 4659031957 2812026006 3345373757 2227446734 5189647036 0864889126 6768754889 1250770836 7527329753 3054043739 9117098508 0803485811 5348882159 4603545076 8363499608 7152180082 1503912600 3775949200 5050176705 7981817066 2275780209 9812485159 5453480733 2151090338 9679954627 1423213108 8200036098 1617017265 7039606278 6061233857 7879970504 3216223209 3882134414 6083502332 6457989981 4003754439 3845655694 4138159699 3538803406 7601760915 2142626886 4511589043 8376552935 5627179064 4770993651 2640140555 0273481931 9853644950 8117829176 5866594940 2703910918 9220381141 8038594933 9344387549 5362763759 3651162625 6632783243 0489956225 1934230668 1108689709 2764227795 2488397786 1961190226 8199661395 8277865351 5739695017 1961980642 0767893382 9126811231 4030645259 9847816848 3823115879 4071363030 3983908097 5169746523 9824088035 8530594369 4527136546 5303307794 4277879303 8858135809 4369255543 1108823598 4312682274 1939117009 7504333363 9835472777 2380240278 3375526385 1104111993 1316416131 5460833240 1975425660 9935941906 3506971948 4764470567 5796167537 6035778065 7248644212 5088141868 9161275384 3611272078 9793120490 1733729500 5650694095 5156456628 4763013540 5547615583 9876107928 3176506820 1064073863 1546874667 9473766453 2596849288 4494825197 2644588234 6992556970 7136853024 3372611866 2420270704 4962637939 5579021662 7234178656 7267594091 4392024913 3151463246 3712944236 5165636807 1933326866 5573270577 1484521455 0412435509 3108710486 4797969878 9281687846 3914065877 0015738820 1023424522 2135264590 9538312270 6544297721 5844491543 8573180019 7026030036 5299414462 3077200129 3200447789 7160944683 6807509157 6968718966 2523850374 0817494638 3867795833 9638790524 1798004851 5897270731 9415309100 8355844151 5848945370 3274810159 0870523935 4791984371 2949926228 9815912598 0581233193 8245598770 9373935385 0819019548 2460831673 6188366092 4026416280 5390023082 7920835291 7746924661 4089799646 4976788085 7187121960 8863901703 1462361410 5410310261 0400327089 9618678412 5367117764 8566137875 1299944236 5790775952 2560334756 8742728298 6214904284 7667772972 5153211557 5668679797 9601559341 2644953971 1574298676 1821786944 7976758227 3955974231 5659730076 5635524854 2938525118 4798455596 5033919558 8014981317 9278744064 0113813938 8485818020 7049866405 4678333620 5481022793 9496072067 1833453135 4188837506 8920279128 1391400605 9246987833 8952396303 8887177526 7249052057 9353132460 1001796378 7850980789 0747866487 5555252875 9059500542

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 ≡ 12 (mod 63)
• Q(1) is not a perfect square: it is ≡ 21 (mod 64)
• Q(2) is not a perfect square: it is ≡ 62 (mod 63)
• Q(3) is not a perfect square: it is ≡ 29 (mod 64)
• Q(4) is not a perfect square: it is ≡ 17 (mod 65)
• Q(5) is not a perfect square: it is ≡ 5 (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 = 3654 3746352781 6282635142 0953283702 4594346580 7488881061 6727754124 4142959394 4890864605 0269451128 5913636641 5262529471 7060222080 3830311021 4642797980 5284576171 8531434730 3878593211 8113184667 2726808198 0553752642 8272132266 3535523975 4038596053 2197201011 9503922395 8390816283 3309241291 2639435163 7645906777 2002610686 8529030404 0144145340 3523477038 6170498211 1696450398 2687398258 7191577835 4833128283 0805136647 6924234096 5336141665 2905919149 8007108625 7961482434 5735434999 8065078722 1781646311 2212678814 1334024606 6916704372 8695510040 1763705157 8497601212 7255113805 4024770837 1338966952 6772438295 0101743468 1973033860 5882407094 0811923474 5557213468 0324133312 3966640267 7639480593 3270679037 0331203541 5520136901 6171089073 4455852385 9871185669 6540464859 7646426930 2112899122 6837034402 6727617762 7347916339 6984904570 5956447855 9380666236 9384806033 4133975507 6691419062 2017326029 4310893972 2914396432 0545173560 3508287844 8874785457 1444292253 3877251342 9827984727 9927572205 1908243654 5431701367 6931045711 0474000949 8841798875 4431505701 1379400895 9193728508 2948945508 1520277501 9618263204 5020519022 0518468630 7447368917 3387799105 5601346664 7592954559 9652998486 3187865744 6454877285 1956909426 0271026987 8621628765 6235875178 8425595811 8377486632 2746489950 0208870097 6919601719 0840250882 2826219646.

With those constraints, the unique continued fraction is: {0, 32, 4, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 8, 1, 8, 1, 1, 1, 17, 1, 5, 2, 1, 2, 4, 1, 12, 59, 1, 22, 1, 4, 7, 2, 3, 7, 1, 1, 2, 4, 22, 3, 1, 4, 23, 1, 13, 2, 3, 1, 1, 1, 1, 1, 1, 37, 3, 2, 3, 4, 11, 1, 1, 1, 1, 1, 6, 3, 1, 1, 1, 1, 2, 117, 3, 1, 2, 2, 2, 1, 11, 1, 2, 37, 1, 2, 1, 1, 2, 4, 4, 1, 1, 1, 1, 2, 2, 3, 1, 4, 3, 1, 2, 2, 6, 9, 2, 1, 6, 1, 1, 9, 1, 1, 9, 1, 2, 2, 15, 1, 5, 3, 3, 2, 2, 2, 2, 2, 25, 41, 1, 22, 2, 3, 6, 2, 1, 1, 27, 3, 2, 2, 1, 1, 2, 1, 4, 3, 3, 1, 2, 1, 3, 1, 21, 1, 98, 2, 2, 9, 2, 2, 2, 1, 9, 1, 4, 7, 1, 2, 14, 16, 3, 2, 14, 1, 72, 5, 1, 4, 4, 1, 36, 1, 2, 3, 1, 6, 1, 2, 1, 4, 1, 17, 1, 2, 1, 1, 2, 2, 1, 2, 11, 1, 6, 6, 1, 1, 1, 1, 5, 24, 5, 3, 2, 7, 1, 37, 4, 2, 1, 4, 2, 2, 1, 19, 1, 1, 45, 1, 2, 1, 5, 4, 3, 2, 15, 130, 1, 15, 1, 2, 1, 2, 8, 2, 1, 1, 1, 1, 1, 2, 1, 4, 5, 1, 2, 10, 1, 36, 1, 2, 1, 36, 1, 5, 3, 1, 13, 1, 1, 1, 2, 3, 3, 2, 2, 7, 3, 1, 4, 2, 2, 12, 2, 3, 12, 1, 6, 1, 1, 5, 1, 74, 1, 20, 3, 1, 2, 8, 1, 1, 1, 1, 1, 8, 2, 9, 5, 1, 1, 10, 1, 34, 3, 1, 13, 2, 2, 1, 2, 1, 1, 1, 91, 3, 105, 4, 4, 2, 3, 2, 10, 7, 1, 6, 4, 2, 1, 16, 1, 10, 1, 1, 3, 7, 6, 5, 5, 1, 1, 7, 1, 1, 2, 5, 10, 1, 1, 4, 1, 10, 2, 46, 3, 3, 2, 213, 9, 2, 4, 1, 16, 5, 2, 40, 2, 1, 1, 1, 17, 2, 1, 1, 9, 3, 1, 1, 1, 3, 1, 6, 4, 1, 6, 1, 1, 1, 1, 1, 1, 3, 11, 2, 3, 1, 4, 3, 6, 1, 2, 1, 23, 2, 1, 2, 1, 1, 17, 1, 1, 1, 3, 5, 5, 45, 1, 1, 4, 1, 1, 7, 1, 2, 2, 3, 4, 1, 1, 3, 2, 1, 2, 2, 8, 6, 1, 2, 4, 1, 54, 2, 2, 1, 3, 6, 1, 4, 1, 1, 3, 5, 2, 1, 1, 1, 7, 2, 2, 9, 1, 1, 2, 1, 2, 19, 2, 1, 2, 1, 6, 1, 11, 20, 1, 1, 2, 1, 2, 1, 12, 25, 2, 5, 1, 1, 3, 72, 2, 2, 1, 2, 4, 1, 1, 2, 1, 1, 11, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 5, 1, 2, 24, 4, 2, 1, 1, 1, 1, 1, 4, 1, 2, 2, 1, 6, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 35, 3, 13, 3, 6, 1, 1, 3, 1, 2, 2, 6, 3, 1, 2, 6, 1, 3, 1, 6, 5, 2, 2, 8, 1, 1, 2, 7, 1, 1, 4, 1, 10, 25, 1, 15, 1, 1, 4, 2, 22, 2, 2, 2, 1, 1, 1, 1, 5, 2, 25, 7, 8, 2, 2, 1, 8, 11, 2, 3, 4, 1, 30, 1, 4, 10, 1, 5, 1, 6, 7, 1, 4, 1, 4, 2, 2, 1, 2, 1, 6, 5, 5, 1, 4, 2, 2, 1, 14, 5, 7, 8, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 2, 13, 1, 2, 1, 1, 4, 6, 2, 2, 2, 1, 3, 1, 1, 7, 2, 1, 1, 9, 1, 1, 1, 2, 2, 1, 1, 2, 5, 1, 21, 1, 1, 2, 3, 3, 1, 1, 221, 7, 1, 1, 9, 1, 18, 1, 5, 2, 3, 3, 4, 1, 3, 1, 1, 18, 2, 3, 3, 1, 1, 1, 2, 1, 1, 19, 2, 2, 1, 1, 2, 1, 2, 2, 2, 15, 2, 1, 1, 1, 1, 2, 1, 2, 118, 5, 1, 2, 2, 23, 9, 2, 2, 2, 2, 1, 6, 1, 9, 2, 1, 1, 8, 15, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 28, 3, 6, 4, 1, 1, 2, 1, 16, 1, 9, 1, 3, 1, 1, 1, 8, 26, 2, 2, 28, 1, 4, 13, 7, 3, 5, 2, 1, 4, 1, 1, 5, 3, 293, 1, 1, 1, 3, 1, 4, 7, 21, 1, 4, 1, 11, 1, 19, 4, 1, 1, 1, 4, 1, 1, 12, 2, 3, 22, 4, 1, 4, 3, 1, 1, 148, 15, 3, 1, 8, 34, 1, 8, 1, 1, 1, 9, 1, 1, 1, 33, 2, 1, 183, 1, 8, 1, 3, 1, 4, 1, 1, 2, 1, 1, 1, 1, 5, 4, 1, 4, 53, 1, 20, 1, 4, 15, 1, 3, 1, 1, 2, 4, 1, 7, 112, 3, 2, 7, 1, 4, 38, 1, 17, 1, 1, 3, 1, 64, 3, 2, 2, 150, 1, 2, 1, 3, 1, 1, 2, 2, 1, 7, 10, 2, 13, 1, 3, 1, 5, 2, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 7, 8, 3, 4, 1, 5, 1, 2, 1, 6, 3, 4, 1, 1, 5, 1, 1, 1, 2, 2, 75, 4, 2, 7, 110, 1, 7, 2, 8, 2, 7, 1, 2, 1, 1, 2, 1, 2, 22, 1, 10, 2, 3, 1, 1, 1, 1, 1, 55, 1, 21, 12, 5, 9, 1, 10, 1, 2, 1, 7, 1, 1, 2, 5, 1, 6, 2, 86, 96, 1, 3, 72, 2, 23, 1, 2, 1, 116, 1, 15, 1, 5, 7, 2, 7, 8, 4, 4, 1, 35, 3, 3, 1, 10, 3, 1, 1, 2, 6, 3, 6, 2, 4, 1, 1, 69, 1, 2, 2, 5, 1, 15, 1, 3, 1, 1, 1, 2, 1, 6, 12, 1, 7, 135, 8, 1, 1, 2, 163, 1, 5, 1, 1, 3, 10, 8, 3, 2, 153, 3, 1, 23, 1, 9, 4, 19, 1, 8, 1, 1, 3, 12, 9, 1, 1, 4, 1, 3, 1, 1, 8, 4, 1, 1, 1, 1, 1, 2, 5, 30, 1, 1, 4, 4, 2, 2, 2, 1, 8, 1, 1, 1, 1, 1, 19, 4, 3, 4, 1, 1, 31, 1, 18, 2, 1, 8, 2, 2, 2, 1, 10, 23, 1, 13, 12, 5, 1, 1, 17, 1, 12, 3, 3, 1, 3, 1, 1, 19, 1, 1, 3, 4, 1, 1, 3, 2, 36, 1, 2, 1, 8, 2, 1, 2, 2, 2, 1, 9, 2, 2, 17, 1, 1, 1, 3, 1, 2, 25, 4, 3, 3, 1, 2, 2, 1, 6, 12, 48, 1, 22, 1, 3, 1, 3, 1, 1, 7, 1, 3, 1, 5, 107, 2, 1, 6, 1, 4, 3, 1, 22, 1, 1, 1, 3, 3, 1, 1, 12, 4, 1, 1, 4, 4, 2, 2, 8, 2, 15, 2, 72, 34, 1, 1, 3, 1, 1, 3, 1, 5, 1, 12, 1, 6, 12, 1, 4, 2, 2, 1, 42, 1, 1, 2, 4, 1, 349, 1, 1, 36, 2, 1, 3, 1, 1, 1, 12, 3, 1, 5, 2, 16, 4, 1, 1, 1, 9, 1, 1, 1, 7, 7, 1, 1, 3, 1, 2, 6, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 5, 3, 25, 1, 3, 1, 3, 1, 24, 1, 2, 3, 1, 1, 1, 1, 1, 4, 2, 1, 13, 3, 1, 1, 1, 2, 20, 1, 69, 3, 1, 7, 3, 1, 1, 2, 1, 1, 1, 5, 2, 10, 1, 2, 1, 1, 2, 3, 11, 1, 1, 18, 1, 6, 7, 1, 5, 26, 1, 4, 7, 60, 4, 1, 23, 1, 1, 7, 1, 1, 1, 15, 2, 1, 3, 2, 1, 1, 1, 69, 7, 1, 1, 1, 14, 4, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 3, 3, 20, 1, 1, 1, 2, 1, 1, 1, 45, 2, 2, 1, 2, 3, 3, 1, 3, 2, 12, 248, 1, 2, 5, 2, 4, 268, 9, 1, 5, 2, 157, 4, 1, 1, 15, 1, 4, 2, 15, 1, 7, 1, 5, 2, 4, 5, 2, 4, 2, 1, 2, 1, 4, 15, 1, 18, 1, 1, 61, 2, 1, 1615, 1, 1, 1, 57, 1, 21, 2, 6, 1, 1, 1, 1, 2, 1, 10, 1, 4, 4, 1, 2, 1, 1, 2, 2, 2, 1, 16, 1, 3, 1, 18, 1, 22, 1, 1, 12, 19, 1, 3, 2, 1, 11, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 21, 3, 1, 2, 2, 1, 1, 1, 62, 67, 2, 1, 2, 8, 1, 5, 21, 1, 2, 2, 57, 3, 3, 8, 1, 4, 2, 3, 3, 4, 1, 2, 20, 2, 2, 3, 2, 1, 54, 3, 12, 2, 3, 4, 1, 6, 2, 1, 3, 16, 4, 5, 24, 2, 8, 13, 1, 4, 1, 73, 1, 2, 3, 1, 2, 1, 7, 1, 1, 1, 3, 4, 1, 1, 4, 2, 7, 23, 1, 1, 2, 1, 1, 1, 1, 2, 6, 14, 2, 7, 1, 3, 1, 1, 4, 1, 1, 3, 8, 2, 3, 2, 3, 4, 1, 2, 2, 1, 7, 2, 2, 4, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 5, 2, 2, 12, 1, 1, 2, 16, 21, 1, 12, 1, 2, 1, 1, 1, 5, 8, 1, 1, 21, 7, 16, 7, 9, 1, 13, 1, 689, 1, 10, 17, 2, 2, 10, 1, 5, 1, 25, 1, 54, 2, 1, 13, 1, 1, 1, 3, 1, 10, 8, 1, 2, 1, 2, 2, 1, 1, 1, 5, 1, 2, 4, 1, 2, 2, 251, 7, 1, 16, 1, 4, 85, 1, 184, 1, 1, 6, 1, 1, 6, 2, 1, 6, 2, 1, 3, 45, 1, 1, 2, 1, 4, 1, 4, 2, 8, 7, 21, 8, 1, 8, 1, 3, 5, 3, 1, 2, 2, 1, 46, 3, 1, 6, 1, 4, 18, 96, 1, 11, 2, 53, 14, 14, 1, 2, 1, 2, 1, 3, 6, 1, 3, 3, 1, 3, 2, 1, 4, 1, 4, 7, 17, 2, 1, 1, 2, 2, 6, 115, 1, 2, 1, 1, 6, 1, 80, 2, 10, 1, 2, 2, 19, 10, 1, 1, 2, 1, 6, 1, 8, 1, 1, 7, 1, 1, 15, 60, 1, 2, 5, 3, 1, 2, 1, 1, 3, 2, 1, 2, 1, 19, 3, 1, 14, 1, 1, 8, 1, 2, 4, 3, 5, 18, 1, 2, 1, 9, 6, 1, 2, 1, 18, 1, 7, 8, 1, 812, 1, 1, 1, 2, 1, 1, 7, 1, 4, 17, 8, 1, 1, 1, 3, 3, 10, 1, 2, 3, 2, 2, 36, 18, 3, 3, 4, 1, 1, 2, 3, 1, 1, 2, 1, 8, 65, 1, 2, 2, 1, 24, 2, 1, 2, 1, 4, 7, 1, 2, 1, 2, 10, 502, 1, 5, 3, 3, 2, 5, 4, 3, 1, 4, 23, 2, 5, 3, 4, 2, 1, 110, 1, 12, 2, 4, 1, 4, 2, 11, 1, 1, 3, 5, 58, 1, 6, 2, 9, 4, 1, 7, 2, 1, 1, 6, 1, 13, 9, 1, 1, 3, 4, 5, 4, 33, 1, 3, 1, 4, 9, 1, 3, 1, 21, 1, 1, 1, 4, 4, 2, 2, 1, 2, 1, 1, 8, 1, 3, 2, 1, 3, 1, 1, 1, 1, 30, 2, 1, 1, 1, 1, 4, 5, 1, 1, 2, 2, 13, 1, 1, 1, 1, 1, 2, 2, 1, 4, 3, 47, 2, 9, 16, 2, 1, 1, 1, 6, 1, 2, 2, 2, 1, 1, 3, 57, 17, 1, 1, 8, 9, 1, 1, 2, 58, 2, 1, 2, 1, 1, 724, 11, 34, 1, 9, 2, 3, 1, 8, 1, 1, 2, 14, 1, 1, 1, 10, 1, 9, 1, 3, 1, 4, 4, 1, 1, 1, 1, 1, 1, 2, 1, 5, 3, 1, 3, 1, 13, 3, 5, 1, 8, 2, 9, 3, 5, 11, 4, 1, 1, 263, 1, 7, 3, 2, 83, 1, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 36, 1, 17, 1, 2, 7, 3, 2, 1, 1, 28, 1, 1, 1, 1, 6, 30, 10, 1, 1, 3, 3, 1, 2, 1, 6, 1, 22, 2, 8, 8, 11, 2, 1, 3, 1, 1, 1, 3, 1, 1, 50, 1, 122, 7, 1, 3, 1, 12, 2, 1, 4, 1, 6, 1, 1, 1025, 5, 9758, 3, 1, 2, 749, 2, 2, 1, 2, 2, 3, 2, 4, 11, 22, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 7, 1, 1, 2, 1, 5, 1, 51, 1, 30, 1, 1, 2, 1, 1, 1, 4, 2, 1, 5, 2, 1, 4, 2, 1, 10, 1, 1, 1, 1, 34, 1, 2, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 7, 1, 27, 2, 19, 1, 2, 31243, 2, 5, 4, 1, 3, 2, 4, 1, 13, 1, 4214, 1, 2, 1, 1, 15, 2, 10, 16, 1, 1, 8, 4, 4, 7, 3, 2, 3, 1, 15, 2, 1, 2, 37, 5, 1, 1, 10, 1, 8, 1, 1, 9, 1, 1, 3, 1, 1, 1, 8, 2, 1, 4, 1, 3, 1, 3, 1, 1, 3, 2, 5, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 5, 1, 3, 5, 7, 1, 14, 108, 2, 6, 1, 1, 1, 7, 1, 1, 9, 1, 1, 1, 1, 7, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 1, 6, 1, 2, 1, 2, 5, 1, 1, 1, 4, 4, 5, 2, 4, 26, 8, 1, 20, 1, 18, 3, 4, 13, 1, 3, 5, 1, 1}, giving these values for u and v:

• u= 22 6995766417 1290945022 5336235512 6124588974 7406513123 0529571221 0107588044 8945609591 1316509553 7204759561 8309527865 1312822973 6493821069 3327785346 1934853659 7581214861 8063141675 6021924381 5698690186 6034644349 7170661670 9531557183 9225061137 8571356751 5783080696 6602378976 5249101765 5975213784 5265833678 9480950172 6183866185 3596084936 7022771196 5311925766 4863516084 5067589527 5385324400 7688550132 9599313714 9904142100 5278111932 8038718685 9863100806 5540416449 6020709551 9308710394 4034251655 8943869028 1856112211 3152527143 7056021295 8501144710 4585273069 1634677542 4198851562 9643542566 7916059330 1069437001 9879166191 6599452091 2737016365 0296113625 5524178569 1885951550 5572090367 1390145176 5341961677 3386252698 6005768972 7359478683 3389738905 4582285312 0964980593 6967117116 7623175112 6090116035 9350758306 4309722513 3778345478 3492052949 5534918217 9258505888 0894579821 4945660922 0789208051 3592742410 5009383554 4791962476 5594885687 0443187203 7195319182 7797598622 7289828165 0626831307 7590989814 4262121006 9366415702 6726177922 5852710682 1912991567 1077110513 9513038112 9513325558 1723105870 9487753077 5908252632 8927912370 7729518575 2812441988 4627697827 2564331395 5575559883 3492136187 5478848307 7772270545 4253399736 5292802561 7262652450 4305542703 8254489171 2877258195 4694188484 9937316304 3729473500
• v= 731 5198119401 9779472557 9120597195 7510212360 3810134004 2038243956 8276560150 5230439113 7486631633 2186642366 0508583717 5447425164 4214595177 8753468244 2694221606 0827879085 0690100921 3095209673 3435549038 6049682786 1965362775 2506324483 2258868891 3073022660 0833936765 8693488659 4176078734 5005414402 3593806213 7994062137 9939309811 2772077938 2518951302 7497888153 2975129980 1197334933 6240177556 1436963704 9032148533 4042162187 0560233390 0791910488 6089253986 6959179517 3409586223 2048410719 7552331744 1201272410 6622605781 0939047711 0771258117 7543180645 7492041974 6547660113 5387400927 6894584453 6432495959 6879052042 2987089124 7751821226 1748382458 2526278372 8980062829 9723090836 3655398779 2490653496 0265904555 0690134892 3739368554 1648185006 2849726192 4435647560 5886171746 0262460409 9564669578 0171000270 5549523676 5366318814 0837652916 9336070311 7987591738 4508719865 1538241123 5222644618 2613020343 9705745242 1061852897 1642077247 7647121010 0552258839 5441008459 1113662842 0106692494 8386891822 1136946980 6900099027 9709217411 4250624232 2483341536 0618532655 6823957269 2439979575 3022221003 9004805340 1680981864 0124138606 2473783883 8540166182 7581015574 8807935048 0281872018 4949030088 6667770735 5028920952 0733916870 7652694033 5956499734 5270574358 7099033377 0788426054 9484008196 4234288685 6224384085 1818346281

… as taking one more term in the continued fraction would give a value of v of 17222 4960821100 1875329345 1641060925 5238676166 9512624449 5654203258 3287058835 7993718036 5656368644 5686476330 0326016702 1302933042 8844789710 3082679240 8559800117 5351195496 8668806670 8774204444 7321917492 2446902879 3474742217 1594539982 2214873779 4106636561 5697766375 4025231410 2158196759 4745313407 7201960598 2541792079 5381405688 7768669934 7364278363 2117760753 9503531102 6748393652 2184283279 5644312578 3248323650 9930718972 7321636486 5725819612 5908273159 2155399804 8283990220 7119577473 6213780572 8584200745 1099505196 6727674488 0951113996 5852642238 7671856809 2883368498 9377841958 9377931486 5906561731 6428839056 8863547993 2899516244 9846462182 1356579297 6065172826 1987873104 9701731837 4328093148 1567820847 7265866925 3019613631 5310099719 6818744044 4249869285 6505483402 4236264536 4095036966 4147358558 4378073033 9101223957 8521955350 7628502734 0632769271 6804123463 8740212338 8094058005 5518594811 4324781410 5280547740 2094302104 6885974156 6237622304 7007225836 5145605824 1375684286 0181832811 8139285553 1306800693 0396250147 9369654988 8205298438 0541881622 9716269330 9652955810 6538874320 5737081869 7922848848 0194694985 8021547724 6505330625 7710838049 9341949188 2258951119 8875105723 7218138618 1287475791 6676255453 5936252898 0779653976 5900693133 8546808135 3382938041 5013257129 5542618613 8364420063 6991300828, which is too large.

We also need to calculate d = floor(c₄·v/F + 0.5) = 119463592 1167011388 6758372224 2746536979 1608618512 3076916301 3149476348 9188911521 9814829372 2758155269 5335247625 6623110634 8740076231 0210470076 7922996462 8084493261 2431962194 3047913621 4203986507 2787027915 3959648774 9649500812 2956181486 8280032546 4549063049 4842723313 7946252677 4426508163 2044675014 9035195925 7756961448 5235788073 7554025283 7108755474 2177447557 4193566210 7040047954 9638163676 7343432471 8312127866 6255670812 3540570700 6704362748 8804914772 6311135324 9117827073 8746464073 8070635013 3481879485 8932237053 6738087143 7555190315 8686035441 2475279168 6397856340 9381448683 5325110144 5186320902 8255690431 3942953630 3919668887 1018699854 0330510942 0950976625 4602293180 1387313179 5928784045 6410622833 7709187462 6942446434 8881442715 1147196299 2290861063 2967727500 2263500534 6957617792 8029470477 3475665347 0708831553 4578469620 5254872210 9160629277 1648056700 3555623911 7997957408 6639938192 0138903581 8835750488 3680184417 5591407564 7776770464 1067452914 8527392160 9991054492 5940762974 6401585887 3402632796 4678085962 3728701611 0631717309 8509337430 8058597682 3737768566 5653431037 4720537588 2076339261 1676465500 9041799294 3631364911 9329542468 2152251713 3757651702 5751960818 2052505111 9498495616 2176315686 5448199361 9611784085 6087408830 1327361712 8133264675 8746504415 8533456744 3582946546 8298227848 0849471831 0485614235 4142151398 7200986939 7223192937 9536133833 7759175676 1261693588 2218700110 3712504381 4626323165 7528312251 0658621434 6884714102 9537424040 8541413713 6954610094 0854926183 3541325171 2975804166 2108855916 3106581204 7050011327 2381135837 4344382206 0038957732

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= +731 5198119401 9779472557 9120597195 7510212360 3810134004 2038243956 8276560150 5230439113 7486631633 2186642366 0508583717 5447425164 4214595177 8753468244 2694221606 0827879085 0690100921 3095209673 3435549038 6049682786 1965362775 2506324483 2258868891 3073022660 0833936765 8693488659 4176078734 5005414402 3593806213 7994062137 9939309811 2772077938 2518951302 7497888153 2975129980 1197334933 6240177556 1436963704 9032148533 4042162187 0560233390 0791910488 6089253986 6959179517 3409586223 2048410719 7552331744 1201272410 6622605781 0939047711 0771258117 7543180645 7492041974 6547660113 5387400927 6894584453 6432495959 6879052042 2987089124 7751821226 1748382458 2526278372 8980062829 9723090836 3655398779 2490653496 0265904555 0690134892 3739368554 1648185006 2849726192 4435647560 5886171746 0262460409 9564669578 0171000270 5549523676 5366318814 0837652916 9336070311 7987591738 4508719865 1538241123 5222644618 2613020343 9705745242 1061852897 1642077247 7647121010 0552258839 5441008459 1113662842 0106692494 8386891822 1136946980 6900099027 9709217411 4250624232 2483341536 0618532655 6823957269 2439979575 3022221003 9004805340 1680981864 0124138606 2473783883 8540166182 7581015574 8807935048 0281872018 4949030088 6667770735 5028920952 0733916870 7652694033 5956499734 5270574358 7099033377 0788426054 9484008196 4234288685 6224384085 1818346281
• z₂= -125951089 6401839539 4789054170 5404864402 3578177042 9799041160 6743565438 3937924594 2376388254 8741827829 7333368475 9610715609 2172763491 4038730203 2166803241 9298042819 8601851162 5586567594 0841273682 9129039537 9869928099 9645812229 6870704761 3818904145 6776356615 1800240928 9301459923 9359805609 9260147050 9118652478 6361031376 4765697084 7866731633 0213161329 9163643142 6773594550 3567750696 9866927349 1188576912 2760602705 6494959046 0921929924 0022875079 0782559294 3056106595 2731287850 4691870149 0310341163 8990989817 9952234269 4671991444 3478197651 6598295861 2477562034 9356477222 4639031084 9809740607 9732146397 1033353154 0913107855 6664609092 4156445342 0479605457 2117640335 7724516092 7004220774 8925238775 4673515355 5694085529 8909917341 0678557843 3793679795 1998262222 6800926533 8167195985 6700662395 7630298462 3109556999 6487846708 0564963680 5701358392 0085801649 3701170600 8081263101 1468172674 7644032389 7944924195 2017091349 0527641250 9558998707 4522508230 9070455084 4797453395 8470755137 6276027620 0098799787 1463142474 7557928230 6141991535 3182558227 3528398724 8512837317 9194380324 8455052490 0841564416 8774554579 8988764215 9792396933 7591368028 8805772085 9637006646 2991225222 7001980156 0700663487 2988005542 2010215522 9962567704 0775594084 6208314578 9252471567 7026113815 1661946792 9670693331 6452896008 0195842932 9063908685 1466175189 4316383445 0210811068 5266037377 5202844125 1064147854 8188794151 1846921524 2534206593 7800896631 8228443975 3726109845 4049528775 3026394291 4672679036 6588808764 4951261429 2145654389 9952541488 5905952781 7537060700 9297319915 7896695371 2634486187 4234563453
• z₃= +38 8554029866 3374911198 3643478830 3216198700 6281984766 6035381368 5761438393 1351573420 0664697714 4846832607 7313202897 3768901780 1127057518 9307600475 5870556939 9766659439 8855331403 8643519267 4235875909 3274101661 5906013246 7525930886 7685528827 1511392847 7271863561 9436959223 7505350976 5850042429 7160300378 5089470879 2911384871 5183851007 0501398508 0044909702 1845689113 7606732822 9448114595 0407237305 4001421378 3282434950 2427458023 6666011424 8557122443 0522124398 5112884917 8589365332 7487698970 5627115179 7401573082 0066908292 4971041885 8729785240 9728912633 9141786568 3694750323 5848656325 8961902672 1979353975 1073714294 4546826804 6970506662 0801390830 0764672603 0113826851 1178127055 9707457513 7423118362 5772071555 8680285148 2633615860 5734938518 0872615009 1502748138 1296909136 9756797911 9011593326 2400207182 6802663933 4745663686 7687119744 6285926094 2999253700 0868751872 3745889187 3207782230 0013513131 0805812775 9938190141 8690130921 9014281352 9839617502 9494184825 7850056580 7306447978 1860120782 6144426831 5185701008 9507766844 7220541522 5611905088 3604227343 5700635786 6286324851 7718940974 0827380588 9656297208 1038080720 2855350277 1618434139 2727000721 5798116683 5934090731 0960938551 5760410484 2194752319 6380397767 6945897420 5278745892 6072574866 6568291104 0757843096 1653898453 9800261063 6454312206 9938979871 3514622899 0103213705 0418701774 5847625935 1548727743 8594865824 0443714664 1998626876 9689893418 4027874565 8215332757 4312627352 9703486933 7709645159 6743609620 7997847760 4196680528 7752242089 6764122861 9893894542 4540487291 2469304380 2250360906 7823986511 7760108728 3816286533 9438895770 8348334837 4560365315 5268978650 5724643539 4329101841 2865352032 9152648670 3576724598 3392508818 0886001465 3447259889 2087531162 3128003778 5283750401 1948004087 6056899940 5579441371 3240672507 2529276769 3500720400 8732193657 8382672974 0528166380 1938679008 5892845193 9484251023 1173272043 7508263379 4474669940 8072682866 6245793813 8306828434 9383981641 8160258183 6855815328 7120661021 2182552355 5718870549 1466127102 9423871811 1288266369 6114531425 4751968247 4957346605 7991823387 6572698902 9726813911 4390605553 1444885624 2205898446 1061123119 7416308785 1180947368 8090425017 2134485374 1368408259 4732942172 2484255976 7834119949 3017636007 4637427867 9374750870 0346107569 2478673408 9113773766 0291952510 9266233806 3266422642 2474900893 4567838346 7701693733 9078346178 3921462535 2697613516 2609907045 6895964967 6414476386 7742295408 1253974961 9554441446 0211058111 7458019377 3570606902 8384427777 6163526051 3245531494 4648187419 2163832953 6590477915 7654717668 1489179190 4403495480 0911106278 7902507933 6897505457 6190777537 9592965911 4860062982 1371595021 7573517697 2759542114 8665242078 6354859855 6639713622 4398344145 6877898924 3417528928 2826869403 7353963039 6441190916 3408274444 7363968214 7536631739 6421949217 7172112026 3899479298 1089389682 1879168965 7821753814 5306291985 3925563278 0053589521 0072380202
• z₄= -119463592 1167011388 6758372224 2746536979 1608618512 3076916301 3149476348 9188911521 9814829372 2758155269 5335247625 6623110634 8740076231 0210470076 7922996462 8084493261 2431962194 3047913621 4203986507 2787027915 3959648774 9649500812 2956181486 8280032546 4549063049 4842723313 7946252677 4426508163 2044675014 9035195925 7756961448 5235788073 7554025283 7108755474 2177447557 4193566210 7040047954 9638163676 7343432471 8312127866 6255670812 3540570700 6704362748 8804914772 6311135324 9117827073 8746464073 8070635013 3481879485 8932237053 6738087143 7555190315 8686035441 2475279168 6397856340 9381448683 5325110144 5186320902 8255690431 3942953630 3919668887 1018699854 0330510942 0950976625 4602293180 1387313179 5928784045 6410622833 7709187462 6942446434 8881442715 1147196299 2290861063 2967727500 2263500534 6957617792 8029470477 3475665347 0708831553 4578469620 5254872210 9160629277 1648056700 3555623911 7997957408 6639938192 0138903581 8835750488 3680184417 5591407564 7776770464 1067452914 8527392160 9991054492 5940762974 6401585887 3402632796 4678085962 3728701611 0631717309 8509337430 8058597682 3737768566 5653431037 4720537588 2076339261 1676465500 9041799294 3631364911 9329542468 2152251713 3757651702 5751960818 2052505111 9498495616 2176315686 5448199361 9611784085 6087408830 1327361712 8133264675 8746504415 8533456744 3582946546 8298227848 0849471831 0485614235 4142151398 7200986939 7223192937 9536133833 7759175676 1261693588 2218700110 3712504381 4626323165 7528312251 0658621434 6884714102 9537424040 8541413713 6954610094 0854926183 3541325171 2975804166 2108855916 3106581204 7050011327 2381135837 4344382206 0038957732

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.

As z₁ is > 0 and z₄ is < 0, we know that P has at least one real root r₁ > 0. This is easily found:

• r₁= 0+ε∈(0,1)

Note that the root is not an integer but actually lies in the interval (r₁,r₁+1).

To see if P has any more real roots, we examine the quadratic derivative P' of P = 3·z₁·x² + 2·z₂·x + z₃, which we express in the usual quadratic form as a·x² + b·x + c

• a= +2194 5594358205 9338417673 7361791587 2530637081 1430402012 6114731870 4829680451 5691317341 2459894899 6559927098 1525751152 6342275493 2643785533 6260404732 8082664818 2483637255 2070302763 9285629020 0306647115 8149048358 5896088325 7518973449 6776606673 9219067980 2501810297 6080465978 2528236203 5016243207 0781418641 3982186413 9817929433 8316233814 7556853908 2493664459 8925389940 3592004800 8720532668 4310891114 7096445600 2126486561 1680700170 2375731465 8267761960 0877538552 0228758669 6145232159 2656995232 3603817231 9867817343 2817143133 2313774353 2629541937 2476125923 9642980340 6162202783 0683753360 9297487879 0637156126 8961267374 3255463678 5245147374 7578835118 6940188489 9169272509 0966196337 7471960488 0797713665 2070404677 1218105662 4944555018 8549178577 3306942681 7658515238 0787381229 8694008734 0513000811 6648571029 6098956442 2512958750 8008210935 3962775215 3526159595 4614723370 5667933854 7839061031 9117235726 3185558691 4926231743 2941363030 1656776518 6323025377 3340988526 0320077484 5160675466 3410840942 0700297083 9127652234 2751872696 7450024608 1855597967 0471871807 7319938725 9066663011 7014416020 5042945592 0372415818 7421351651 5620498548 2743046724 6423805144 0845616055 4847090266 0003312206 5086762856 2201750612 2958082100 7869499203 5811723076 1297100131 2365278164 8452024589 2702866056 8673152255 5455038843
• b= -251902179 2803679078 9578108341 0809728804 7156354085 9598082321 3487130876 7875849188 4752776509 7483655659 4666736951 9221431218 4345526982 8077460406 4333606483 8596085639 7203702325 1173135188 1682547365 8258079075 9739856199 9291624459 3741409522 7637808291 3552713230 3600481857 8602919847 8719611219 8520294101 8237304957 2722062752 9531394169 5733463266 0426322659 8327286285 3547189100 7135501393 9733854698 2377153824 5521205411 2989918092 1843859848 0045750158 1565118588 6112213190 5462575700 9383740298 0620682327 7981979635 9904468538 9343982888 6956395303 3196591722 4955124069 8712954444 9278062169 9619481215 9464292794 2066706308 1826215711 3329218184 8312890684 0959210914 4235280671 5449032185 4008441549 7850477550 9347030711 1388171059 7819834682 1357115686 7587359590 3996524445 3601853067 6334391971 3401324791 5260596924 6219113999 2975693416 1129927361 1402716784 0171603298 7402341201 6162526202 2936345349 5288064779 5889848390 4034182698 1055282501 9117997414 9045016461 8140910168 9594906791 6941510275 2552055240 0197599574 2926284949 5115856461 2283983070 6365116454 7056797449 7025674635 8388760649 6910104980 1683128833 7549109159 7977528431 9584793867 5182736057 7611544171 9274013292 5982450445 4003960312 1401326974 5976011084 4020431045 9925135408 1551188169 2416629157 8504943135 4052227630 3323893585 9341386663 2905792016 0391685865 8127817370 2932350378 8632766890 0421622137 0532074755 0405688250 2128295709 6377588302 3693843048 5068413187 5601793263 6456887950 7452219690 8099057550 6052788582 9345358073 3177617528 9902522858 4291308779 9905082977 1811905563 5074121401 8594639831 5793390742 5268972374 8469126906
• c= +38 8554029866 3374911198 3643478830 3216198700 6281984766 6035381368 5761438393 1351573420 0664697714 4846832607 7313202897 3768901780 1127057518 9307600475 5870556939 9766659439 8855331403 8643519267 4235875909 3274101661 5906013246 7525930886 7685528827 1511392847 7271863561 9436959223 7505350976 5850042429 7160300378 5089470879 2911384871 5183851007 0501398508 0044909702 1845689113 7606732822 9448114595 0407237305 4001421378 3282434950 2427458023 6666011424 8557122443 0522124398 5112884917 8589365332 7487698970 5627115179 7401573082 0066908292 4971041885 8729785240 9728912633 9141786568 3694750323 5848656325 8961902672 1979353975 1073714294 4546826804 6970506662 0801390830 0764672603 0113826851 1178127055 9707457513 7423118362 5772071555 8680285148 2633615860 5734938518 0872615009 1502748138 1296909136 9756797911 9011593326 2400207182 6802663933 4745663686 7687119744 6285926094 2999253700 0868751872 3745889187 3207782230 0013513131 0805812775 9938190141 8690130921 9014281352 9839617502 9494184825 7850056580 7306447978 1860120782 6144426831 5185701008 9507766844 7220541522 5611905088 3604227343 5700635786 6286324851 7718940974 0827380588 9656297208 1038080720 2855350277 1618434139 2727000721 5798116683 5934090731 0960938551 5760410484 2194752319 6380397767 6945897420 5278745892 6072574866 6568291104 0757843096 1653898453 9800261063 6454312206 9938979871 3514622899 0103213705 0418701774 5847625935 1548727743 8594865824 0443714664 1998626876 9689893418 4027874565 8215332757 4312627352 9703486933 7709645159 6743609620 7997847760 4196680528 7752242089 6764122861 9893894542 4540487291 2469304380 2250360906 7823986511 7760108728 3816286533 9438895770 8348334837 4560365315 5268978650 5724643539 4329101841 2865352032 9152648670 3576724598 3392508818 0886001465 3447259889 2087531162 3128003778 5283750401 1948004087 6056899940 5579441371 3240672507 2529276769 3500720400 8732193657 8382672974 0528166380 1938679008 5892845193 9484251023 1173272043 7508263379 4474669940 8072682866 6245793813 8306828434 9383981641 8160258183 6855815328 7120661021 2182552355 5718870549 1466127102 9423871811 1288266369 6114531425 4751968247 4957346605 7991823387 6572698902 9726813911 4390605553 1444885624 2205898446 1061123119 7416308785 1180947368 8090425017 2134485374 1368408259 4732942172 2484255976 7834119949 3017636007 4637427867 9374750870 0346107569 2478673408 9113773766 0291952510 9266233806 3266422642 2474900893 4567838346 7701693733 9078346178 3921462535 2697613516 2609907045 6895964967 6414476386 7742295408 1253974961 9554441446 0211058111 7458019377 3570606902 8384427777 6163526051 3245531494 4648187419 2163832953 6590477915 7654717668 1489179190 4403495480 0911106278 7902507933 6897505457 6190777537 9592965911 4860062982 1371595021 7573517697 2759542114 8665242078 6354859855 6639713622 4398344145 6877898924 3417528928 2826869403 7353963039 6441190916 3408274444 7363968214 7536631739 6421949217 7172112026 3899479298 1089389682 1879168965 7821753814 5306291985 3925563278 0053589521 0072380202

As b²-4·a·c is negative, P' has no real roots. Therefore, P has no turning points and is monotonic, implying that r₁ is the only real root of P.

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