Primality Certificate for (3164^3061-1)/3163

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 30.865746% factorization of N-1:

We need the product F of all the prime factors from this partial factorization:

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) = 30.865746%

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 = 34 suffices.

Express N in base F

Let N = c₃·F³ + c₂·F² + c₁·F + 1. Let c₄ = c₃·F+c₂.

• c₁= 87433 4727489860 5767262777 2783721343 5870261638 0499162465 9912126571 5712145550 0417926940 7019261691 6641184498 5242752730 8113581642 9465288531 1053459880 3689554159 2901336992 2839959304 4182691832 7117161761 2588018376 8337506146 5329238112 9173042271 6775949573 7384300658 7285026339 6759838061 6054227601 3206180503 7390010370 0813336037 0050306566 0910871816 9655839909 2242163278 7020306550 9132070220 8433348779 9046583781 8505685117 2859556384 9293821240 4973054470 1940003927 1754262691 4636116742 0576277586 8574617058 7515979477 6548229223 1084364240 5070667558 2016879889 0873919446 4034125393 2296870554 8345111795 1023813138 8607887262 8490803354 4070746312 8256744694 9209782633 9656818147 4395892415 9059823669 3891295202 8153488593 2010107170 3486749431 7725604742 5680691446 8840472786 8113512055 9367601255 5735672675 5749562432 4676898379 4136286950 8543171525 8661678013 5111318038 8440610664 6366200334 8041570758 8787598118 3449668796 4480608405 0152314786 8111395822 0423322617 9987000815 5463276395 1799649146 8422739296 8436193163 4954517609 5768657958 3995264048 2253439846 8000449691 6908100444 9400954359 0032019809 8109763207 3667823586 3447728541 1791000753 0901642429 0416321493 0770899906 8485493408 0472130654 8990316662 8802057758 0638385851 6091767052 1703131118 0982772403 7974030283 8165274101 5673554698 9589115893 6322840620 5757714374 4953279028 6271690641 6683903968 7274685860 7684472440 9851140774 4956704441 2111284300 5470720429 5114981048 0686786455 9602752213 8602501736 6706119601 8960345970 8428280718 1606921061 6963969238 7912836782 2569997612 4325983788 7531479035 4095697730 5644983216 8318860097 4655740445 1741306162 7661440797 5513255586 0803794471 5945821924 7529068038 4926369766 4037482089 6730513692 9633884514 3108368907 7552586652 3446474923 5231127309 2956628766 2530100119 0212873949 6705396579 6361724069 1801918265 2420660962 9924651636 3306558295 8003726381 4613291469 6858572674 1759022752 5414700969 5668692417 6432600064 9284788978 5185748339 1602380171 2146993156 7399588718 6798723936 8154325947 4753276177 1943897472 5853661065 7237656267 1595143510 4026088690 6834966051 2482565068 0282916122 2346754168 5998923327 8077293315 9190385646 8082326672 0846661600 6556799797 5218682217 1910693176 1532325659 1918359699 9003622486 3140115130 1396704767 0235177068 4836457631 5554669581 9949737746 4455481117 5487257017 6820590355 3997124566 9634509119 6966760943 5591458555 2160356347 0761517218 0277593906 8748187899 3633621444 4484395857 4445536863 2406548891 6494962269 5303914704 5599170149 6564587959 1946427197 6946278676 4597563239 7596486035 6540677957 0039204142 0828346566 5184925623 6644567880 6516386647 2171152448 6301636184 8014193485 2446761144 7961429485 4811707124 7419731870 6894327538 6208713980 5335858735 0282382664 1198257401 1670486723 1507487902 5600932260 7097041146 3892069106 6395565872 0474903086 9517624339 4582249965 7667463894 3263365037 1600182979 6916014951 1343900236 4048084553 8651144128 4885573717 0962472869 4796938007 6236093349 1478354214 0056934217 9626968484 1376481036 8918941155 4196427242 9368648962 8709466412 0332392341 1515069518 0991249463 5091608706 9222959861 5968413149 5541394142 3015115273 8169735141 9447298235 8530691122 5770707002 2491059960 8254783483 5862145346 9162244235 9093076983 2478423768 1836588802 0578164134 3617032640 9300715759 9763479165 3940430723 4228050725 5842133382 7877235904 7735798379 9816924935 1948501998 8245732117 6734317097 8957703325 1031462500 3115841620 0256643726 2013795889 4346509838 1630801697 7853477796 7706596837 5384432906 2954062311 5286028214 0994931272 4583189987
• c₂= 731247 4321938168 7455469256 3699257943 2252656592 6614845912 7104439526 5510650026 8254359064 7056959219 8305449813 9308457965 8309595734 2081814372 6048220270 7375471727 3790411019 3160752688 1456201976 6501290998 8040394805 3389032410 3744809533 5684618298 2047626574 0543768079 9834962613 7179446513 7036674791 5726481004 0624042821 2787790954 0005573510 8500285355 9007994814 7590979539 3655173077 8110625806 0268966548 6485041051 1329478866 4951594097 1802923849 9896210954 1621742622 4267682224 5214963180 5781090286 7710636932 7275324387 4831295211 1063745340 9979822007 5924449629 8182609175 2803921519 1032786955 6775839403 0257196784 5731806142 4886670307 5524473956 4688109494 5942542310 4305334069 6485660110 1068846828 9908057025 4999295865 8630578118 2476726055 0559946245 6939554793 5724732511 1923018077 9882634033 3484263751 1586860129 5554959342 8524483776 2291995651 9531677657 7747672250 2273674802 3301965549 3717211209 5607252571 8511773332 3663748636 6165124341 0560842635 1674950623 9406832348 9853868881 2171171488 5279634104 6148107792 2383217561 0561518805 5149467522 8948063200 7254478725 7483706353 6334822522 3076297027 0393982314 6748108317 9793278592 8715299710 9039842153 5308006919 0850866240 6115306573 7711221805 9726866356 2267060778 7018546339 6146669017 2076316278 5745004705 4597387992 6346572358 3695444546 9557783743 6708825837 2734399175 0086497305 1271506558 9505607510 1597839511 8829699659 1545075297 3478764344 9014028646 6307057550 7526221451 2067339293 0838985825 8233286563 0805402012 8152293906 8732868897 9812494383 7821737644 7275643500 1600453797 9096930154 8614546003 8681976586 6065487596 5926475053 5219156102 5041438859 9648049350 4910410185 5188584786 7592756745 4207669812 0040093827 9133787375 0073197433 2548481632 8668247603 6158723736 4581482351 5198858134 2156170064 0758851197 8166755649 4073867471 1529380380 8999951149 4967663777 7579482682 6500873583 4518169668 1667265946 8035614546 1281439979 8288382236 6395820389 0680770195 4820966756 8356881379 4271439240 0377376598 0652554841 9097377352 6991730696 8013689652 5753402883 5121258440 6681866109 8233509170 9119410516 6396166231 7766929717 6646107967 0007229858 0317385263 0885770289 9862525397 4028385618 5351945109 5256486221 1925877928 5156691912 2784889795 0207843642 4843874071 5114729897 2388883246 8624949908 7426586320 1422394339 4705271090 4664497865 3722756219 4376973853 9477308858 3575648232 6794972359 4129712016 2945536173 2712525900 6230190024 3520437489 2594494653 2124724063 2099624956 8009255849 9019381386 2319906307 4831830651 8431140094 6449866272 9484542949 0370305150 8655840858 5878857502 9009365694 3332472011 7286447715 9085399237 3983503427 1003865703 4390677162 3081003629 4636546275 6657149618 8605406185 2885186838 4138218336 9022553280 6888477080 2772090479 0659149242 3341509745 9546891890 7458672239 3263855690 6914473469 2721394614 9833710103 3076613307 0415144990 5349291431 2408798318 6767509576 7753690325 0841696890 5166915155 6467946707 6350003279 7923505026 0919158130 2809652318 6012077099 5462415505 9267467628 7463435147 4886877125 2930438288 9045059686 7261132828 2041101433 0424099945 2257561254 5268217920 7001188943 8568225252 7630585999 7136482124 3955698021 2003426337 4388796927 0242552400 4399501281 7006341673 5270888964 2778437940 9718110961 2075205237 9225592301 5927511617 1028955848 0224521715 4255923124 8211425664 1572400761 2473783793 2438073858 9098605339 2538407376 4388321338 8731719532 0739991627 7460413094 6354917881 3680439062 6952414974 8048155232 9108760075 0597941251 8311482650 5570351708 9852364597 3156340718 9822560318 5216999195 1859783226
• c₃= 775 3907675257 8960767376 1505856607 0308886057 0805062499 5176953152 7989399182 5811673043 5756555889 1994718331 6566619409 9521613029 4670429665 5666145470 8814005806 8177194411 8744135140 3397734160 5965982723 6981353163 1491393251 5984551175 3033197247 4788318757 8441613504 2705471997 7381902030 9753537053 1212438285 8239706472 6826532869 0777141367 6215981877 6000633785 1937195012 3214269879 3938924924 5502932197 3334858716 8276658037 6911539823 4708409797 2581995568 4023295334 2656052029 7090442996 3454979084 4266372401 2161673325 7056668449 1123126139 1822425508 9063873676 4883192727 3020148310 8409417716 2086324359 4681970234 4302247845 8271441049 1684449502 8341375568 9374471295 8959763392 9754236650 7915882886 0160384261 0106200585 4657987852 4649772457 2456025303 2532329225 7482761988 9873099124 9445706982 5611861937 3069718981 8105914202 2098905810
• c₄= 682753340 2597273595 9808170990 2631657049 8116319793 1218022684 8228850046 3503751909 5702167216 0461044845 9652576329 1307942423 1457580415 4056363861 6162095024 3651083032 8449263364 8423873871 4897752879 9620085663 9923556262 3300230890 7158364204 5315684684 0447429862 5051881501 7994672701 2002735954 1047180953 1949343173 2047535044 9969376290 6969961666 1718128008 5516977805 2247386037 5063623808 3278029516 3654375591 8843858090 7233113517 1098813555 6107584644 4096978731 1692049188 4020358351 9077804367 7678322551 0755171739 7016699647 8221794820 8775417984 3058860594 3948019844 6747889633 7623582587 3646229518 4347770370 2713601914 8364433913 7235408496 3646098420 4820301651 1558430451 4042565617 5914155689 5011637482 6509095799 7670959986 2725161038 1756955133 8897462891 1748846121 4422236016 3212253580 4516746753 1319633073 9038154815 9750068941 9887839226 9596238833 2538660126 9131448737 7854589354 1597689519 9643416232 8941771913 9853743825 6965346462 3842870125 2945417538 4725169260 6611380499 1942862897 5704889472 7880307804 8747650843 3292653225 3098740919 0041362471 5447881238 4231094723 3523243260 9330171025 9700559728 3067648425 6551328176 5680835290 8321891686 5817786552 8675558368 7259660686 4366385452 1344944565 7934054201 2185466361 9442833446 1519911540 1512853205 2554899835 6890180485 1040992576 2858049481 0333297025 6971812790 8571979576 4861671458 5756985490 1550830155 9711401189 0039242896 8268181448 0750217484 4067540999 2816343941 3802340974 9127332652 8567877446 9061243590 6736879410 5665074724 0291774826 4263265379 0560494796 1816852066 4984950354 3529111989 9260767314 0115684733 9389743258 5418353741 7029881159 3363096263 3519557768 8844682092 3259981469 6738952238 9329729291 7732424583 8571836148 2411395207 3123201108 2079461236 7163319005 7574235272 9198917919 7346732532 9390968876 4904864199 2811706257 9412711515 3801983861 0373716757 2091790926 6594035411 6480869549 7681943357 3170724525 0184178865 1257375393 5154212437 4425955327 6637185142 6641056518 0609508982 1148718965 0509586634 9486082951 0831820192 9982116426 9836606936 7760448084 8648202716 1167873296 7676042547 5683963600 4378728944 5246659603 9219401707 2786530916 6645940966 7590764625 1750064612 1752867775 1025235487 7724485650 5794012348 2618992655 7999412656 7613730957 7593289120 8975953805 8654242816 7140291352 0153236469 1692962727 4729774308 2073281275 2024370579 6101491189 1292260207 3077336118 9572775053 0516857390 0757262723 7742676052 2547807024 4457376802 6262629808 8454210112 8319229418 7851959587 8682085730 8887493532 7327454848 6151817982 7684070557 6325009838 1934868923 6570879587 4856158606 8140690020 3382233637 5327270300 3045057596 3215164684 0724134477 8710955219 0122342672 4203056784 5870279410 6307868744 7521227581 9976601513 1256099678 1481759472 2035144790 0478217875 0660954426 7143038162 3044848233 0632072172 9603743386 0682198691 1716890275 8676578334 7755802009 6355870056 3061533741 6311265928 2591004334 5467706949 0627151127 3187284139 3795776878 9318954973 1910919421 7063004797 4387670405 1502636878 5779237459 5530823008 1910088838 8801514923 5100354014 0711715576 2688255289 8962775428 5504463426 4367721680 7801092760 9559501438 1072560767 4173167546 0026190465 3027084553 7435265266 8352852506 7642089135 9603339160 4348753720 9816885351 3392280157 8442927931 1228425715 5043417305 6832649288 3729293552 4012090323 8707619699 5455624014 0258465423 7512279892 7199052081 6312554127 0080876362 7153078793 0637271247 2980018365 4510548744 1538320779 9195325691 4028096458 2297165977 0847044528 4659255156 1058846475 0713104818 6092595988 3517792539 1329761226 9292068788 4552014545 5754517662 3561890518 3845971828 1412204074 9618419936 1483696736 1700213329 8425674059 0578677547 4748824229 9786500917 4544745875 8273946781 8923945911 7113658245 2277198418 5156506342 0787193498 2224238716 3745394291 5982791920 5633023555 5944216607 7539305865 9289683509 4379985108 4929482168 4007691617 8973742132 6495108878 9774900571 6958563376 2566125797 7083192772 7856523883 9739072702 6863937581 0635379682 6336455715 5454386521 8769028959 1627165167 7319450110 0227231489 5837379498 8015534432 2575426356 8634053931 4061807453 8453834447 5200972668 6567013649 9524128294 6931894538 0691977195 7441118035 4406328694 0990009848 4929468442 3018529553 2330239696 2395309981 7369070613 0441282650 3913450708 1807421559 3539098969 0151994815 1535921921 2515741907 1529644194 9790061033 9379473730 9628328143 8759823019 2621268094 1336452226

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 ≡ 38 (mod 63)
• Q(1) is not a perfect square: it is ≡ 61 (mod 64)
• Q(2) is not a perfect square: it is ≡ 13 (mod 65)
• Q(3) is not a perfect square: it is ≡ 37 (mod 64)
• Q(4) is not a perfect square: it is ≡ 54 (mod 63)
• Q(5) is not a perfect square: it is ≡ 29 (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 = 3369855 5970235720 1569808765 3481767072 0176068117 9186000908 4211357039 3956208205 3482714057 0115496216 7590091323 7549460165 4131635766 8979289496 9178509456 4843899433 9957783371 8033296273 6505507029 3431624186 4490979057 3218251718 5992847665 8956430947 6318374238 9921014663 6882123751 0008551392 1249974108 5455540654 7961332330 8943855941 8333771046 6314322905 0451346118 3069256756 6874942371 9087534732 6067890783 9519420560 9774086265 6781994886 6983175660 6065052201 6290287954 0599062729 3737758822 7775734104 3228318094 4187970385 7491640100 5084376997 1579429415 4016521224 2070600263 9890228424 6576154819 9945564367 7570185475 1474212059 0848152042 6391293125 1753971306 1195030329 0428695470 5568600767 1544375187 2157910518 3802274569 8364540254 6859082590 8393987754 6701406471 1556743446 5844051662 3290484062 3864522619 0249976558 1817935989 6115349662 2210560524 6732576029 3320959956 4806746568 0858574732 7677316549 9741915825 8474144969 7685307605 9964682784 9310782497 9195427775 9199189927 2235998754 0899762056 3779762479 5013759948 6612247454 8453238296 0247173836 7228690703 6300089092 9659331453 6879498691 1832818856 9414089430 1662093724 8322768033 6510269609 1096422434 0339190218 1157625551 8586583801 2069677074 6207373553 5337772446 6657886658 5461871145 3144232831 2893100003 7247604620 8379676180 9752391611 5480162892 5234735666 2964601329.

With those constraints, the unique continued fraction is: {0, 10, 14, 8, 1, 1, 10, 26, 1, 1, 5, 2, 15, 2, 2, 1, 22, 5, 2, 2, 1, 4, 4, 1, 10, 1, 6, 1, 5, 5, 11, 5, 1, 1, 4, 1, 876, 2, 5, 1, 3, 25, 1, 1, 1, 63, 1, 1, 1, 5, 1, 1, 3, 3, 1, 2, 3, 1, 5, 1, 1, 1, 1, 13, 1, 9, 1, 2, 1, 3, 2, 1, 1, 1, 2, 3, 1, 204, 1, 1, 5, 1, 2, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2, 11, 1, 10, 2, 2, 2, 3, 1, 7, 2, 4, 17, 2, 1, 26, 1, 6, 3, 3, 1, 1, 8, 1, 2, 8, 1, 220, 1, 1, 9, 1, 4, 1, 1, 1, 61, 1, 21, 15, 13, 2, 6, 1, 3, 26, 6, 1, 4, 3, 4, 23, 1, 2, 2, 47, 2, 2, 98, 4, 1, 1, 5, 1, 1, 5, 1, 5, 2, 1, 2, 2, 2, 1, 1, 8, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 4, 1, 13, 2, 2, 5, 12, 1, 2, 1, 1, 7, 1, 1, 1, 2, 1, 4, 1, 4, 5, 4, 2, 1, 1, 5, 1, 1, 2, 20, 1, 2, 1, 1, 1, 12, 1, 2, 2, 4, 3, 17, 6, 2, 1, 3, 42, 1, 7, 77, 1, 2, 5, 10, 5, 2, 1, 1, 3, 1, 1, 1, 5, 9, 1, 41, 11, 2, 1, 30, 1, 8, 6, 1, 1, 3, 1, 1, 2, 1, 1, 4, 2, 2, 1, 3, 3, 73, 1, 6, 7, 1, 1, 1, 1, 1, 1, 2, 2, 9, 3, 9, 1, 1, 1, 1, 5, 1, 96, 6, 1, 2, 3, 1, 3, 1, 2, 14, 2, 2, 1, 1, 1, 2, 2, 12, 1, 2, 14, 1, 1, 1, 1, 1, 3, 46, 3, 14, 3, 1, 1, 17, 1, 1, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 64, 4, 23, 2, 2, 5, 5, 1, 2, 1, 1, 1, 1, 1, 2, 9, 24, 9, 1, 2, 1, 3, 2, 1, 11, 1, 4, 1, 1, 2, 3, 1, 12, 1, 1, 4, 1, 3, 2, 10, 2, 1, 1, 2, 1, 1, 2, 1, 6, 10, 1, 2, 1, 1, 2, 2, 1, 3, 50, 1, 3, 1, 9, 1, 1, 1, 8, 1, 1, 25, 1, 4, 29, 1, 2, 5, 13, 1, 4, 1, 4, 1, 10, 1, 1, 1, 1, 1, 6, 1, 1, 1, 3, 7, 3, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 17, 1, 7, 1, 11, 1, 1, 1, 3, 1, 1, 2, 2, 6, 2, 18, 1, 11, 2, 1, 2, 76, 6, 2, 1, 1, 7, 56, 24, 1, 15, 1, 1, 2, 11, 1, 7, 6, 4, 1, 41, 1, 2, 798, 9, 9, 1, 6, 3, 1, 7, 1, 2, 1, 4, 16, 2, 3, 2, 3, 6, 1, 11, 1, 2, 1, 2, 2, 1, 1, 2, 2, 3, 4, 2, 2, 1, 3, 1, 7, 1, 12, 1, 5, 1, 4, 2, 1, 75, 4, 2, 1, 2, 1, 4, 1, 4, 1, 3, 9, 8, 2, 1, 3, 1, 4, 1, 1, 8, 1, 6, 1, 11, 1, 1, 2, 1, 1, 1, 6, 7, 3, 1, 7, 1, 6, 2, 1, 5, 3, 1, 16, 1, 2, 1, 39, 2, 1, 4, 1, 6, 9, 1, 7, 3, 1, 3, 1, 8, 5, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 12, 1, 3, 5, 1, 1, 9, 3, 84, 4, 1, 36, 15, 4, 1, 3, 1, 4, 1, 41, 2, 2, 32, 2, 1, 2, 2, 1, 1, 3, 1, 1, 11, 1, 3, 1, 1, 1, 8, 2, 6, 2, 15, 1, 1, 1, 1, 69, 1, 2, 200, 1, 1, 2, 4, 8, 1, 1, 1, 2, 1, 1, 6, 5, 1, 4, 1, 6, 1, 9, 1, 4, 10, 41, 1, 2, 1, 5, 2, 1, 1, 59, 2, 5, 1, 3, 3, 11, 1, 1521, 4, 9, 44, 1, 22, 2, 1, 1, 2, 6, 2, 12, 2, 1, 1, 1, 6, 1, 6, 1, 1, 1, 4, 4, 1, 8, 1, 194, 1, 2, 1, 2, 3, 1, 2, 4, 1, 6, 1, 3, 2, 2, 3, 1, 5, 3, 2, 7, 4, 2, 2, 2, 3, 1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 1, 5, 42, 5, 1, 1, 2, 1, 5, 3, 1, 12, 1, 27, 1, 3, 1, 4, 1, 68, 2, 10, 2, 1, 1, 1, 2, 3, 1, 29, 8, 67, 2, 2, 2, 4, 5, 1, 1, 2, 1, 6, 2, 6, 1, 5, 1, 1, 3, 1, 1, 4, 3, 7, 2, 1, 2, 1, 4, 1, 8, 1, 1, 1, 1, 27, 1, 1, 2, 4, 1, 1, 4, 1, 1, 3, 1, 2, 8, 1, 1, 1, 58, 1, 4, 1, 7, 1, 1, 1, 10, 1, 1, 3, 15, 1, 1, 8, 1, 48, 4, 8, 4, 3, 9, 1, 8, 1, 9, 1, 55, 1, 3, 1, 4, 3, 3, 13, 1, 9, 1, 3, 2, 6, 4, 13, 1, 17, 2, 4, 3, 3, 1, 1, 1, 52, 1, 14, 1, 2, 6, 2, 3, 1, 3, 2, 2, 1, 74, 20, 1, 5, 58, 2, 1, 1, 1, 4, 1, 11, 1, 47, 1, 1, 4, 2, 1, 3, 2, 2, 2, 3, 10, 1, 1, 1, 51, 1, 1, 2, 1, 17, 6, 1, 20, 2, 9, 1, 1, 1, 1, 21, 11, 26, 1, 77, 2, 45, 1, 12, 8, 2, 1, 1, 1, 1, 9, 4, 3, 1, 2, 7, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 2, 10, 1, 12, 16, 2, 1, 1, 2, 1, 2, 5, 10, 1, 4, 4, 1, 1, 2, 1, 2, 3, 2, 3, 5, 1, 2, 2, 1, 2, 3, 2, 1, 2, 72, 1, 1, 11, 1, 2, 1, 18, 2, 1, 1, 2, 3, 4, 2, 104, 1, 3, 3, 3, 2, 1, 1, 1, 4, 3, 1, 3, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 1, 3, 18, 26, 1, 6, 1, 2, 1, 3, 2, 3, 2, 2, 3, 15, 1, 47, 1, 1, 1, 3, 1, 13, 1, 4, 9, 1, 3, 1, 10, 1, 4, 2, 7, 1, 15, 1, 1, 1, 1, 11, 2, 3, 6, 7, 1, 1, 1, 1, 3, 8, 1, 82, 1, 2, 1, 5, 19, 1, 4, 4, 26, 1, 4, 3, 56, 2, 1, 14, 3, 14, 5, 2, 1, 2, 2, 9, 10, 1, 2, 25, 2, 2, 2, 8, 2, 1, 1, 1, 1, 8, 2, 25, 1, 2, 2, 1, 2, 2, 3, 2, 7, 9, 2, 1, 15, 1, 3, 1, 2, 1, 8, 6, 6, 1, 2, 7, 1, 2, 1, 3, 3, 1, 1, 2, 2, 14, 2, 3, 2, 2, 1, 1, 15, 1, 44, 1, 2, 1, 1, 1, 5, 2, 1, 2, 1, 42, 1, 9, 1, 4, 25, 4, 2, 1, 4, 16, 6, 3, 1, 1, 4, 7, 75, 1, 2, 37, 1, 1, 2, 1, 13, 1, 1, 1, 3, 388, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 64, 1, 2, 1, 5, 2, 2, 4, 9, 1, 4, 1, 29, 2, 1, 1, 6, 1, 5, 1, 1, 4, 1, 19, 2, 2, 3, 1, 5, 1, 5, 1, 3, 1, 1, 2, 1, 1, 17, 23, 4, 1, 2, 10, 9, 4, 3, 2, 2, 2, 5, 2, 1, 2, 1, 1, 5, 1, 2, 3, 3, 6, 1, 8, 1, 7, 1, 2, 4, 1, 6, 1, 118, 1, 21, 1, 16, 1, 1, 4, 1, 3, 5, 2, 2, 2, 1, 1, 2, 4, 1, 3, 3, 1, 7, 5, 2, 1, 2, 1, 103, 1, 9, 19, 7, 2, 1, 2, 3, 4, 5, 1, 4, 4, 2, 1, 1, 5, 1, 1, 1, 1, 7, 3, 4, 1, 3, 1, 30, 5, 6, 11, 1, 18, 1, 4, 2, 4, 1, 1, 3, 1, 3, 2, 33, 355, 1, 1, 2, 5, 1, 1, 2, 1, 19, 1, 2, 1, 7, 1, 2, 2, 2, 1, 5, 3, 1, 2, 1, 4, 1, 5, 1, 25, 17, 50, 5, 2, 3, 1, 27, 1, 1, 1, 11, 1, 5, 11, 1, 3, 1, 2, 5, 1, 5, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 5, 1, 1, 8, 2, 2, 8, 2, 2, 1, 18, 26, 19, 17, 1, 17, 2, 1, 1, 2, 1, 1, 4, 1, 2, 1, 3, 3, 1, 3, 1, 1, 3, 1, 2, 3, 3, 2, 2, 2, 1, 3, 13, 2, 3, 2, 7, 1, 2, 1, 1, 1, 2, 1, 3, 2, 1, 12, 1, 4, 2, 5, 1, 10, 7, 1, 2, 1, 1, 5, 1, 207, 1, 4, 2, 1, 1, 1, 2, 3, 3, 1, 8, 13, 1, 6, 1, 11, 6, 11, 20, 11, 1, 28, 3, 2, 1, 3, 1, 5, 3, 1, 1, 1, 11, 3, 3, 1, 4, 1, 8, 1, 24, 1, 5, 2, 1, 1, 5, 1, 8, 1, 2, 3, 3, 1, 3, 2, 3, 1, 5, 6, 2, 1, 24, 1, 2, 1, 1, 1, 5, 1, 16, 2, 12, 4, 1, 2, 1, 1, 2, 2, 1, 1, 2, 16, 1, 1, 1, 12, 1, 24, 12, 2, 1, 15, 1, 1, 3, 4, 6, 65, 1, 2, 6, 1, 25, 1, 38, 1, 1, 3, 1, 2, 2, 1082, 1, 35, 12, 2, 1, 9, 1, 1, 1, 5, 6, 1, 13, 18, 1, 1, 3, 1, 2, 9, 2, 1, 122, 2, 16, 1, 1, 3, 1, 1, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 24, 1, 3, 2, 3, 1, 6, 1, 3, 1, 8, 3, 1, 2, 1, 1, 2, 1, 1, 5, 1, 2, 3, 1, 5, 1, 2, 172, 8, 1, 2, 7, 2, 1, 1, 2, 3, 12, 2, 1, 1, 21, 3, 10, 2, 1, 21, 1, 11, 2, 1, 1, 3, 2, 1, 1, 11, 5, 4, 6, 2, 1, 1, 18, 1, 1, 3, 7, 8, 1, 1, 3, 1, 3, 2, 2, 55, 1, 1, 8, 8, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 5, 1, 24, 18, 3, 1, 8, 1, 2, 2, 9, 2, 1, 3, 1, 1, 1, 2, 1, 1, 8, 2, 5, 2, 3, 1, 7, 2, 10, 1, 3, 1, 5, 7, 1, 4, 1, 1, 1, 100, 1, 1, 5, 1, 2, 5, 3, 1, 154, 2, 2, 1, 1, 2, 1, 43, 1, 11, 1, 8, 5, 1, 2, 1, 1, 4, 32, 1, 10, 2, 5, 1, 1, 1, 2, 1, 31, 4, 9, 1, 16, 4091, 3, 2, 1, 2, 2, 2, 4, 3, 3, 5, 5, 10, 8, 2, 1, 1, 1, 5, 7, 1, 12, 3, 25, 1, 9, 46, 2, 8, 1, 7, 1, 7, 1, 1, 13, 1, 1, 1, 6, 1, 1, 2, 2, 2, 2, 2, 7, 1, 9, 4, 1, 1, 1, 1, 3, 3, 1, 9, 1, 8, 2, 6, 1, 6, 1, 19, 1, 1, 1, 19, 2, 1, 54, 1, 1, 2, 8, 1, 2, 2, 4, 16, 1, 1, 9, 36, 1, 10, 16, 1, 2, 1, 3, 25, 4, 1, 561, 6, 1, 1, 1, 1, 4, 1, 4, 4, 1, 1, 1, 3, 146, 2, 1, 1, 5, 2, 1, 1, 2, 1, 5, 3, 1, 3, 1, 4, 1, 11, 3, 2, 1, 5, 9, 4, 2, 15, 4, 1, 1, 1, 3, 1, 7, 1, 4, 1, 2, 9, 3, 3, 1, 1, 1, 1, 1, 11, 2, 26, 1, 2, 1, 5, 1, 1, 1, 1, 1, 1, 1, 6, 1, 8, 2, 1, 3, 3, 6, 1, 1, 1, 1, 1, 3, 5, 1, 9, 1, 3, 3, 1, 1, 2, 1, 271, 1, 4, 2, 1, 2, 3, 2, 2, 8, 20, 2, 1, 36, 1, 1, 74, 2, 3, 3, 4, 1, 3, 4, 2, 2, 8, 1, 8, 2, 18, 1, 6, 1, 284, 1, 21, 2, 1, 3, 2, 1, 2, 40, 1, 10, 2, 1, 2, 10, 1, 5, 1, 1, 2, 2, 1, 4, 5, 2, 16, 1, 2, 1, 5, 2, 1, 1, 51, 11, 2, 1, 13, 1, 2, 4, 3, 2, 1, 1, 1, 2, 2, 1, 1, 4, 3, 2, 4, 1, 1, 7, 13, 1, 1, 4, 3, 1, 1, 5, 3, 1, 2, 18, 1, 5, 2, 3, 3, 2, 3, 5, 1, 2, 5, 5, 1, 6, 1, 1, 6, 2, 2, 57, 1, 1, 1, 3, 1, 1, 1, 19, 2, 1, 3, 1, 6, 1, 11, 1, 1, 2, 1, 2, 1, 10, 19, 1, 6, 4, 1, 4, 1, 3, 4, 1, 1, 3, 1, 3, 1, 1, 1, 1, 4, 1, 93, 2, 2, 19, 31, 9, 1, 2, 3, 1, 2, 1, 15, 9, 3, 1, 3, 1, 19, 3, 1, 1, 6, 4, 22, 1, 1, 2, 1, 1, 4, 26, 1, 2, 1, 1, 1, 18, 2, 1, 1, 4, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 9, 1, 1, 2, 3, 8, 1, 1, 1, 1, 1, 4, 1, 21, 1, 19, 2, 5, 5, 1, 25, 2, 1, 4, 1, 5, 1, 69, 1, 2, 1, 2, 12, 1, 3, 4, 16, 12, 2, 2, 2, 7, 1, 1, 3, 207, 2, 8, 2, 1, 9, 4, 1, 2, 1, 1, 1, 1, 1, 14, 28, 1, 3, 7, 10, 12, 2, 1, 1, 2, 7, 1, 3, 1, 10, 7, 1, 1, 3, 1, 1, 1, 3, 35, 6, 4, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 2, 23, 4, 2, 367, 1, 3, 1, 1, 4, 1, 3, 2, 1, 1, 2, 4, 1, 1, 3, 2, 2, 1, 1, 2, 1, 3, 4, 2, 5, 2, 4724, 1, 1, 14, 2, 1, 1, 69, 2, 2, 3, 17, 6, 7, 1, 5, 6, 1, 3, 3, 1, 3, 8, 1, 2, 1, 1, 1, 2, 8, 2, 1, 1, 1, 2, 3, 1, 3, 8, 1, 3, 1, 2, 2, 3, 1, 1, 5, 1, 2, 2, 2, 2, 5, 1, 5, 1, 21, 4, 37, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 32, 2, 1, 1, 2, 1, 25, 1, 1, 72, 1, 1, 1, 1, 1, 30, 1, 1, 13, 1, 7, 1, 1, 3, 1, 2, 9, 1, 2, 1, 1, 10, 1, 1, 2, 7, 30, 15, 1, 3, 3, 1, 8, 1, 3, 1, 3, 1, 1, 1, 99, 5, 1, 1, 2, 1}, giving these values for u and v:

• u= 197064 6715562135 8957201211 1032961642 4754627565 1529194202 8162192482 6552611521 7143768881 5532962394 1987912192 3095499622 4589796895 4920791278 9973990794 0836287687 0624393979 5141712207 7215468174 6324796756 3010525875 5182050849 1734773124 0719641963 7481785160 8085401177 1131190521 2508691372 1678604835 3268271484 6052522403 4251208940 1813992450 0126911461 8080294107 5646952972 3682069337 5592589637 1785227666 8460588432 7823173123 7370952811 8955743218 1127121567 1214492014 0492964806 1303931812 3782941618 8906463249 7614476344 8669516896 2631077324 6393370570 6249981220 5699145599 4323941296 8956585021 0889389455 5146860729 3425214553 9201877207 3304360927 5099633319 1449602607 2566791478 9449546199 3866884337 3503462759 9642138373 2035531560 6965918365 8167554410 0186108045 4315347328 5039739650 2949993353 1702001145 3584524053 9270414510 8849486155 5461472216 3695840534 0042880974 9126897690 1654615479 7509028869 7014622163 1417707333 7327257290 5936099540 1408107744 1167559071 1991697843 3000258631 6199710943 7299598872 4928037412 8150615235 6342647508 5697820869 8861052275 9099775638 8900175280 5679843609 6020996178 8194230969 2134374455 1580737634 4537722293 1887366377 3941658483 4590472323 3529027106 4976777294 3458582437 0275601778 3979680635 0009038917 5595286059 4494332430 7998648175 9417143264 3556591613 2006007155 9762647877 0183348426
• v= 1984605 7869040001 9318434078 7794825304 1329452582 8626266967 7223935932 5086361187 4643831316 3189228216 9149581421 7762252834 9046131427 9978825115 1045088918 5227912524 7391204384 5033858923 8089333288 0915426687 7153833542 7789306463 3825424291 3438107810 8878646598 3590016109 2500472328 4023344867 7188696510 6084387080 6452492732 3177244569 6251814981 2828719345 2025469884 0337088720 4996529544 7772310838 0550467994 1139345218 7245930897 6184671498 6857875318 3624315534 0594135378 6810920695 9610330975 2676033642 4721576084 9846284272 3798818847 2170648429 6333399217 1459218226 9596896871 7563469093 0470045361 4468939929 6658483208 1389568518 7471407325 0480014910 9982164576 7408120345 0816936547 2386912027 2272677770 7639318691 3971449846 3698884762 0204397917 8785046690 7515823421 1809598788 1639659480 7326899352 4656983160 6301499935 3208105758 5113163479 5898644234 6487395302 2136090139 5344687619 6919153088 4110139651 6875421823 4320732577 0985020905 3472757823 1444086262 5805678444 4213355684 8765055796 5186115728 1738723403 8985848928 2257108570 0284326373 1561306422 1332300654 2891968303 1059456245 5496996812 6335994701 9453156088 8892016597 3490664952 7653714102 0370029468 1098186830 6896879788 1220665071 3175819155 9540764522 8646193772 2477560075 4521734270 1526705535 8811644303 3787084950 9241761777 2504352081 4695363723 0479684589 2586482745

We also need to calculate d = floor(c₄·v/F + 0.5) = 1538845004 3436163632 4468516765 6108183855 7170097791 0349106369 5569716898 8670514338 2074812426 8448305402 0043414223 9926887696 0976830742 1007858259 4579671951 7957780527 5982227396 4676463541 7970984459 7149467121 0943136744 5393349528 5039805923 6430095836 7909893348 9331343520 3562306541 9303236549 5142676925 1264056587 1075512960 4188243377 8729830266 3432021272 4948691138 0732230144 9213924934 5612492437 3942622151 5390519857 1533283577 9103550837 7193817072 2432073121 8826483019 8094042547 3933341588 7023026755 2704100769 7929516765 2660796516 4192861257 0069013815 0141373280 4590085716 9804356789 7971352431 6858949154 1344505541 6831181977 1926481868 9710941840 0031551463 1273521438 4392984323 5936743729 6539522769 6080741963 3019457684 1716649468 3305183444 2926518421 1303028589 8126875664 5438509069 9587312763 0472806353 5668808487 9492988637 3339553012 1357915221 7161654333 7575089263 6210846325 1217723858 7990318859 0545595161 8701218488 0138340050 8182926284 5787687370 0149049330 7253726295 2560304247 8732238194 4007909215 9499702996 2821934037 8386945063 1513737282 0962848180 8643261662 7040286429 7620089461 5811722287 9578153763 0912944455 5499900350 9130085077 0744632559 5198147302 4865166740 6251868248 0168290595 2961231498 7478643555 0557531472 4923195008 1545275674 8252079911 5826727360 3569414199 3610885011 9075218419 7706354887 7518905053 5378288608 5057954096 3759573502 0585249171 9301971371 2157860893 0747480981 4050019868 5514672763 4149983803 8594999569 4972020667 5259869906 2823692657 0764630793 8570004129 9704664568 1968675439 7859429539 6170475514 7301946518 9600294336 8101485863 0454097980 9891745249 6369419320 7186626680 3135153956 4169994706 9895523900 4766494960 5143115675 1944564327 8671708132 7012164019 9854817743 4121396503 0662231336 2439604089 1888733418 9534969674 7669309352 6530231716 7836502691 4129827370 6221761411 1275582425 2015684846 1896991102 6365355923 0129602838 6213592258 2178699488 3196376889 5348180479 5383538598 7304704775 2701828592 9850287046 9501974531 8033337020 2480097098 1407649430 3539784985 4883900620 7319168783 6095607605 4116079766 2890248660 6073859027 5353173269 6382454384 8822727427 1169807198 5037791783 5528496547 5207246160 4643768333 7320824440

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= +1984605 7869040001 9318434078 7794825304 1329452582 8626266967 7223935932 5086361187 4643831316 3189228216 9149581421 7762252834 9046131427 9978825115 1045088918 5227912524 7391204384 5033858923 8089333288 0915426687 7153833542 7789306463 3825424291 3438107810 8878646598 3590016109 2500472328 4023344867 7188696510 6084387080 6452492732 3177244569 6251814981 2828719345 2025469884 0337088720 4996529544 7772310838 0550467994 1139345218 7245930897 6184671498 6857875318 3624315534 0594135378 6810920695 9610330975 2676033642 4721576084 9846284272 3798818847 2170648429 6333399217 1459218226 9596896871 7563469093 0470045361 4468939929 6658483208 1389568518 7471407325 0480014910 9982164576 7408120345 0816936547 2386912027 2272677770 7639318691 3971449846 3698884762 0204397917 8785046690 7515823421 1809598788 1639659480 7326899352 4656983160 6301499935 3208105758 5113163479 5898644234 6487395302 2136090139 5344687619 6919153088 4110139651 6875421823 4320732577 0985020905 3472757823 1444086262 5805678444 4213355684 8765055796 5186115728 1738723403 8985848928 2257108570 0284326373 1561306422 1332300654 2891968303 1059456245 5496996812 6335994701 9453156088 8892016597 3490664952 7653714102 0370029468 1098186830 6896879788 1220665071 3175819155 9540764522 8646193772 2477560075 4521734270 1526705535 8811644303 3787084950 9241761777 2504352081 4695363723 0479684589 2586482745
• z₂= - 1795609455 7127143080 1791930448 8173296348 7461883327 1989806376 4477477084 6201953728 5138293856 4064213097 7347412751 8071771228 4440673604 3027706489 4274226820 2954162780 6223149741 9099600079 0931342262 5001816819 8810250014 7726212103 4236459623 1416333351 4294687037 1056463606 6437380732 0716687493 1403028164 9773092588 7257718076 8584820754 3591358710 2280663223 7917121093 1474287860 7761923228 9048453321 5596168837 4450157065 9301449365 2704522363 0758744709 7298152394 4641214884 0398807037 4915517472 2796596700 4613337761 8468783748 1106448200 7756692471 2251464565 2013480410 7536284409 4743550497 6838468985 2112537822 2078030702 1168223709 7898867647 2250632175 3457712660 0449646278 7533327332 9958619968 0263494138 8507673448 4619050118 9200690454 4424662622 2490537699 3251772314 3437432263 5128837911 5586665281 0591808575 3886503751 9612916241 3669481526 8858716675 4813736130 0240687796 1618417394 0662961609 8501377605 2377962212 8043574156 9278084700 4946657294 3803453907 7789982507 4313723668 5890012340 7723195422 7592647504 8701933187 8640901617 6606904495 1791969515 9984860754 7245434859 4015397655 3543794703 2391661702 9249593593 9569299762 3237366579 5371599354 0070990898 0289018726 9467876198 7425651303 4919660468 9780541794 0114354361 7365950083 0286915997 5819803466 8562318093 3933273266 2954943115 6977238268 7739482683 7587717372 7066216871 3230569393 5961530840 7488917058 6960032523 7300022786 7530500633 5365238386 0361645808 9268175030 0035005357 0399159191 1213320248 8237089077 5421362914 3607559926 7084841894 9250778715 1030938204 1802751157 2591757201 2406253568 6845390661 1779749057 6095859019 0077648416 8920196063 0078063738 3560362191 4859554376 3638961532 0598906773 3484278290 1271691833 5059544237 4441859264 2383223137 8615900024 2465715034 4877763987 4419417289 1800828668 0388788447 3305241233 9758883514 0298590320 5654602381 0186680409 9498272793 6777583796 0379975715 7712481275 7458754632 6390241728 3095080310 8984580571 2332915781 9145102125 8857215166 5012199529 7694685978 3286988537 9773121873 2490621620 7025184574 2641749535 7596853111 3429479018 4079690098 3232608269 9520861245 5157512489 5575005818 9439633263 5220668242 4741144524 9912741064 7944341999 9827937597 4595266915
• z₃= -427636 1740517949 6364268322 0863980283 8764402892 4201907595 0803672180 8365640065 1031875894 8679866178 7633129574 8037979382 1856074433 5323842468 0441984759 3388839088 2899997150 3167159117 6367934272 7503476989 5198092805 8239979653 1237165041 5064389746 1134072815 1309379393 7513075710 9868257614 9150081356 2894596016 8539422135 8430258435 0948270661 1598886920 7218358635 6136006104 4912124162 0306657451 8573548700 6483473862 5262250298 6281805785 7918662458 3502321650 8241255879 3941001525 6112637733 8635380168 6421096678 7239272688 7555230059 5641863491 0960498853 1623950300 1354608907 4118126092 2830350281 4167831630 1594623022 1241078703 7719431772 1356944442 2559309712 8481014329 5438746230 4590844283 8764066348 2008671772 7389004888 5955708560 2062708072 7086384575 2430269696 6206234093 2437654681 2617586399 0475658211 2460554114 7290038856 1981115836 7953139897 0474538037 3137137555 2614156549 7595757197 6184073398 1843423830 6359495129 4896894110 0123418598 0418274538 2339645545 6400720014 0843031201 7033469462 0312706725 4872656423 6042408454 4504041881 7078346727 0785342255 1262907793 2094450374 5600686998 1815958434 1613260774 5788522144 9823821380 5129320110 5887597670 1157691063 5453610490 1096944438 6930349128 2413276967 4468161417 7342344729 1962482691 1865615958 7227268774 7142959846 2715982280 2495346251 2508472037 9482917038 2459971171 1541466667 4808353125 2719791997 9600795629 6422242776 0154245247 7439579628 6075646461 8575897380 8784344483 6580636565 4942250443 2244902412 8850199312 4666760763 2696959290 1504554371 2911988549 9295902946 0999838353 6276896824 3700528946 2428102991 9381969936 7187998647 5455522696 7508769386 8017171342 9063208227 8395374291 8704161302 5838468275 2436166580 9993283547 9304694594 0569582482 7909570825 5192711600 7766089149 0794524734 7105376045 8450493355 5849189357 5614990596 2449459636 8402655286 5161653341 4027303873 5036121088 5023438679 0876919184 4751552925 3404756542 3305292037 3328789087 5170560353 2013648626 2975918661 7833716383 2543526592 4203321578 9650191521 2006299198 1747670657 1516740024 5225931023 1655911255 0893248008 0895826077 9298803996 1357565898 4779055580 6962017862 1671044011 7100863278 9373333389 6501425132 0427504195 9709951287 7699231800 7542590803 0620342069 6780714300 9168111525 8633261311 6712472025 0875850277 8683398776 6210323947 4956756415 1267934250 2024756870 3741747401 8719675446 8999990467 2984313682 4988264760 2039764536 1704866680 2375946394 8395534154 1399475394 7475345626 7056684840 5268107867 3964298039 6227593980 7129422008 5795143661 2276663637 7473812529 7296401671 1760136636 1308744524 4387714702 4810455916 8710161335 4750792314 7682926145 5916700228 5861720451 7740473113 7915036511 0028847452 8393674833 0452246959 4853395115 5008926652 6416283252 2179870340 2536963575 2574564239 2237887965 1163412949 1352605919 1385753773 4827856334 2428797870 7222013747 7254189716 1339087149 3897014786 5152405566 6856845322 8046582755 7182745566 0923165487 0068675740 2655898346 2592930380 9971529857 0237144975 0650462968 7652560213 1324859823 5957415074 4075194554 7200796809 0690433749 4486189123 8789240717 8124206212 8059229197 0295960398 4063443323 5498227983 9734623371 4176636982 7170918675 4336885045 1770152263 1711587222 0719645609 4247119862 5511668451 3640176123 1842769004 7168095677 0049411510 4548332397 0634575492 6530942875 5427586812 2361754612 7599475429 6953722427 9045942739 6096846280 7006575363 4609792981 5648011234 6778875879 8080414787 1442493971 0717965784 8560818355 8946049942 9377727946 6962753377 7745271777 5822846036 5284407687 3087436315 2186313977 4607367204
• z₄= - 1538845004 3436163632 4468516765 6108183855 7170097791 0349106369 5569716898 8670514338 2074812426 8448305402 0043414223 9926887696 0976830742 1007858259 4579671951 7957780527 5982227396 4676463541 7970984459 7149467121 0943136744 5393349528 5039805923 6430095836 7909893348 9331343520 3562306541 9303236549 5142676925 1264056587 1075512960 4188243377 8729830266 3432021272 4948691138 0732230144 9213924934 5612492437 3942622151 5390519857 1533283577 9103550837 7193817072 2432073121 8826483019 8094042547 3933341588 7023026755 2704100769 7929516765 2660796516 4192861257 0069013815 0141373280 4590085716 9804356789 7971352431 6858949154 1344505541 6831181977 1926481868 9710941840 0031551463 1273521438 4392984323 5936743729 6539522769 6080741963 3019457684 1716649468 3305183444 2926518421 1303028589 8126875664 5438509069 9587312763 0472806353 5668808487 9492988637 3339553012 1357915221 7161654333 7575089263 6210846325 1217723858 7990318859 0545595161 8701218488 0138340050 8182926284 5787687370 0149049330 7253726295 2560304247 8732238194 4007909215 9499702996 2821934037 8386945063 1513737282 0962848180 8643261662 7040286429 7620089461 5811722287 9578153763 0912944455 5499900350 9130085077 0744632559 5198147302 4865166740 6251868248 0168290595 2961231498 7478643555 0557531472 4923195008 1545275674 8252079911 5826727360 3569414199 3610885011 9075218419 7706354887 7518905053 5378288608 5057954096 3759573502 0585249171 9301971371 2157860893 0747480981 4050019868 5514672763 4149983803 8594999569 4972020667 5259869906 2823692657 0764630793 8570004129 9704664568 1968675439 7859429539 6170475514 7301946518 9600294336 8101485863 0454097980 9891745249 6369419320 7186626680 3135153956 4169994706 9895523900 4766494960 5143115675 1944564327 8671708132 7012164019 9854817743 4121396503 0662231336 2439604089 1888733418 9534969674 7669309352 6530231716 7836502691 4129827370 6221761411 1275582425 2015684846 1896991102 6365355923 0129602838 6213592258 2178699488 3196376889 5348180479 5383538598 7304704775 2701828592 9850287046 9501974531 8033337020 2480097098 1407649430 3539784985 4883900620 7319168783 6095607605 4116079766 2890248660 6073859027 5353173269 6382454384 8822727427 1169807198 5037791783 5528496547 5207246160 4643768333 7320824440

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.

The real roots of P are:

• -1+ε∈(0,1)
  (Root is not an integer)
  (Root is negative)
• -46419 4607501616 1609834870 2801532442 8622473847 7199544555 4533035258 6829377118 4298574160 8700246101 9544161834 5431249582 1998639499 9837802671 4437410567 2957743377 4076176412 2518009688 7982884288 9373551739 5104118370 1975890520 1092569627 6695528940 2166817009 1367975736 7427672643 2455403001 3958198043 2512215187 8813402928 1783660225 8963393637 7318739689 0757040161 5916569044 3150741891 9751670368 1050015073 6455742630 9838576747 2040526156 6992762080 1094716531 9705348470 7718578359 0076060090 2809897483 7064962433 5973140559 2706574288 2336003551 6493601335 0728572234 2455887203 0295561106 0154414911 1852320903 3838475661 1606282363 1168242402 7638401061 2463700063 1548273356 0848436607 8930577564 9188879249 5799198203 6714728382 7985079135 8593813701 3983846569 8740747402 4788722968 2665080680 9815675500 0110990392 6101156744 3195858278 8882208465 3830433808 1873084592 7567251723 9366078627 0894302838 4088615567 4899830821 8954629060 8820973918 4907357882 7039619877 3814228791 5511541157 6644121079 8137560731 8284205524 4716785847 5256677021 5520332228 0090553414 7302297487 7744155087 4749698261+ε∈(0,1)
  (Root is not an integer)
  (Root is negative)
• +46419 4607501616 1609834870 2801532442 8622473847 7199544555 4533035258 6829377118 4298574160 8700246101 9544161834 5431249582 1998639499 9837802671 4437410567 2957743377 4076176412 2518009688 7982884288 9373551739 5104118370 1975890520 1092569627 6695529844 9855116426 4151817627 0112923784 6123785322 2630559321 9752277713 4111105100 4716316094 5131868336 7849433220 0203228516 3023552968 6527192728 7179108178 8699120824 6639943235 3314184456 0070481340 6634020604 5812350356 3608774869 1954755553 3394732293 1030419781 9179340913 7017294857 2793604845 5638735229 0538510477 3691359153 2349164524 9249770618 9588007946 2227156038 8223490827 7403369454 1424411818 4934429388 9521469082 1910438759 5061291600 1523238178 4895066872 6113942554 2922215387 2851821425 1355462765 7268046607 0099331556 5325028832 5762900469 7305886105 0794869272 5683746240 8006816877 3699935679 2100236284 1593051768 8729116966 0647351940 0915786198 9884068533 1510497926 3347622890 3165846674 2635636207 8620518612 3690971665 9514721468 4358679433 1106755105 1618080418 6762448680 6638450808 7665428697 3873834644 4352327364 9178925935 5284716508+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.