Primality Certificate for (3048^3121-1)/3047

This certificate uses a theorem of Konyagin, Pomerance and Broadhurst to prove an integer N prime by making use of a partial prime factorization of N²-1.

Factorizing N-1

As N is a Generalized Repunit, we make use of the algebraic factorization of N-1 to arrive at the following 29.971874% 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) = 29.971874%

Factorizing N+1

We have the following 0.153629% factorization of N+1:

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

We set R' = (N+1)/F'. Note that GCD(F',R')=1 and Log(F')/Log(N) = 0.153629%

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 = 473 suffices.

Finding a Lucas Series

Next, we find a discriminant Δ that is not a perfect square (mod N). We choose a pair of integers P and Q such that P²-4·Q = Δ. Let α and β be the roots of the quadratic f(x) = x²-P·x+Q. With α and β, we build Lucas series U and V. We then verify that U_N+1≡0 (mod N) and, for each prime factor q of F', GCD(U_(N+1)/q,N)=1. In this case, P=4, Q=-205375, Δ=236 suffice.

Ensure GCD(F,F')=1

As they stand, both F and F' have a common factor of 2. One of them has a single instance and the other has at least two. To ensure that GCD(F,F')=1 while making the minimum reduction in our factorizations, we divide out the single 2 from F' and multiply it into R' .

Ensure (F^(1/3))/6 > F'-1

log₁₀(LHS)=1395.607318, log₁₀(RHS)=16.398593

Ensure F'-1 > 3.N/(F^(10/3))

log₁₀(LHS)=16.398593, log₁₀(RHS)=0.477569

Express N in base F

Let N = c₃·F³ + c₂·F² + c₁·F + 1. Let c₄ = c₃·F+c₂. Let t = c₄ mod F'.

• c₁= 68680379 2989269214 5171708781 5820292160 7766189391 4678318751 8339107056 3130963599 5087348854 0837552531 4805223714 2969238925 1177218961 7394660604 9010175864 4578490583 0903380856 9492404850 6243623557 0463107220 4421531477 1828926281 4458487312 3671455633 1033004795 6061450307 8448116868 1814932304 1820692045 3613555037 8676265217 1394817808 5735459692 7979236185 5587700419 2508626459 6661638044 0501635075 0930142267 8742896764 0873476832 1156473896 2838059779 7806733206 1647845565 5897802234 8188786997 3271520520 4441254929 8776954810 6926906034 6795670440 8590397688 1065097003 5929218257 1796606451 1032015788 5690634263 5916368867 9341165017 0846185191 9857979955 4126706763 7432635052 6315560326 5481983651 7492013567 8317380565 5561462105 5551255963 0035057077 1954758716 6620993161 6818282357 7806721813 3089085158 9337199307 8602626686 1110803943 1654860355 3466740869 4229756375 9187632650 0126548119 0397035445 4121838801 9606055949 0435986986 9106704245 9203278753 8028066573 9101658305 0417247105 5424834928 8493298805 5086075376 8567347126 6977442816 4361828312 6469939578 0202573406 8018325375 6809652457 6254035647 9493532453 5221320478 9136717514 8515159459 2213874624 3416039873 3584659004 0183520293 6474816625 5759527035 9902659194 3626520395 2805716586 1337619970 4076430350 7229153599 0242671353 6516284333 2467702566 6155330590 9615968837 3575601529 2799372232 8163086747 5231819079 0931311218 4489151070 9035615048 5343488918 7699838639 6747393086 8195487839 3509650320 6847767559 7450101748 9200479546 8705161975 3748907676 9421308172 2312409259 2588447855 2105368124 8270943602 8561941011 7239481169 8893848903 5406692072 0862075355 5814741408 4084468434 7179417014 6249845101 4824011847 6062548693 3998832615 2649076893 6152241328 1061323052 9353546020 7700069679 1295614139 7603820689 3151235141 7920201067 7817850782 2414380218 3946869230 2485997715 2344045855 2929249864 1945299772 7228518670 8547789608 8778859805 3534535203 8247713590 6160311285 6289494625 2091206355 5765820661 3464384196 4578797051 1266987107 1474719727 1387892564 3358263021 9547457980 5894427445 7980212129 5162866804 2732705593 3020418481 4677422167 3520409882 8991004550 3779203195 3220913231 6902913095 4435118995 6913878099 8767025683 5387464099 0965076743 8139159169 8187590494 3642567111 3446126636 0863520175 9038812998 8678070399 1351475499 8989099630 1364236612 7703075104 2381292947 6084577535 7368623426 1920808016 8768958403 8480978859 5744440346 6343333260 2511661603 3534792395 9688843934 6835793564 7426963528 5033207630 3484869415 7722073760 0983434720 8326906837 7057311348 4091148566 9227423565 4504937558 7514315614 1369943759 6587834045 5377958573 1137879283 8168965136 6400749934 5522988665 4428079139 6209848732 7823657504 7381232005 4868217496 7748372287 4717149548 4703779636 5786183420 0439648675 6127251235 8432634043 3532853824 4045208349 3745412532 6664736241 1145392831 2599727242 9440946797 1384505660 4952746504 6890718302 9770189646 6490839313 5266800498 0710764258 5505924015 5558456304 8768669195 4025632360 0280849297 4755739763 8862855038 1494104910 9850286862 5801715343 7072811843 0694366485 2024394423 1137257779 4158772920 2238785472 5303196093 8950676753 5188227546 4439172829 6924122521 7825907222 1779337046 4792300956 7020290858 4656572823 1520914497 3220268458 8822036787 4868965221 2317114658 6140966604 7630030060 0280606268 3803895081 3186731670 8010959504 0734548629 8127426869 5537308092 5302451714 0431647242 4821184731 3727842414 5426757177 3969229791 2648992436 0821771261 8982850307 2473210254 4130492886 0812508549
• c₂= 51215596 3572875771 0356632385 0084142393 6151989745 9055221531 8956267188 5844120290 6472442074 6409237455 5977109491 0526093749 5156995369 2830456384 1621781719 7566901684 7603009726 2220128913 2629089352 6529454069 7538832952 8825341417 6314022227 2978878884 6498651985 9332800489 0876732731 9910015626 5762581061 8272103511 0996935388 1615208294 0929328009 1296263437 2568339046 6953865921 9268332981 0987791557 9958563047 1486781432 3778357914 5143288316 8834058753 5140887960 8822853003 5650397714 3619479571 7858901288 7013713644 4855320838 2757590256 3819142344 6008268208 7923320007 7971646734 3946144855 4093262454 5437038110 1186541575 1109825022 7515228565 3762754997 9211754594 9446844500 2830278516 3481639970 6176736973 4284211461 4618445387 3288384353 3611649823 4790138733 8091845756 1763500064 8438247226 0805043909 5533054614 6594317882 9361925919 4154834664 1474679142 1330590934 2820940372 8569480506 9091626681 0860123285 0077683880 9310325003 1628893449 8417984576 0867882536 7245123284 4560632244 1867767745 7414927721 9477001574 4816270763 5340576030 3958441874 8788562021 1260623823 0428666292 2621812839 6859508727 9648519870 7525148132 7334262633 9878458378 3553856124 2361848902 2923827566 1186578469 7231911320 2789948823 5377083305 2593962871 0386436658 8974945724 5447131174 5824274981 6694322169 1986385090 6247272920 9920681175 3807021800 7997955851 5336094902 4950529532 3228225046 8905225559 6290031896 8363373543 7794523525 3240122135 2018274848 5623279065 6062268983 3619800111 2050812204 1808622839 5330278518 0097301425 8339674209 9841022722 5569323591 8935782794 4536819904 2815261568 4463538530 1272518870 2804586409 3934393638 2614212951 9392836327 7791141436 6270075663 3594669322 8242899438 1821204390 6953645190 6956393606 3203033037 9576153942 7842439879 7514900341 9469356532 5107809704 8264116985 8658145612 4883348888 1920354680 6708950001 1979417558 3162010974 6019600771 5085357894 7161279023 5145671025 4802968357 7518664784 2564267506 2866604344 8003662699 0017388284 2486089156 7601168858 2889776751 4868503219 5413545357 0619813241 1681328638 1857218236 5436864928 8125010602 8235052553 1542874637 4999150355 3140701041 5417696915 0318290727 2257808263 4244182733 9769023424 1142861325 8809031284 6956602715 8131015975 2207237784 0959604830 1054825706 3844160811 4596765764 8700279383 1133734654 8697784057 9376089376 8358154226 5490427917 1464075862 4636976025 0621162365 9955329375 4586429285 2658696178 0884142897 2414610582 0406550285 8193495384 3097444391 3247416394 7194123180 7067244902 7707371465 9872558970 3347650975 4938723860 8654075265 0345061770 1471232433 8116958587 1719505044 7506087678 8629189970 8071673811 0806165712 1201500905 2213963225 7304051496 8395061660 7075471946 4790143297 8906782171 1176382346 9918058180 3068304538 2116436003 1598555340 2268857090 7644117674 9149794920 2085372860 6294969310 5123520754 0922404797 4550693228 2923040632 0526673269 9363175362 3606619552 2934500677 2337090199 6504318559 7090929037 1222977622 5782773082 9858188051 3673925182 9012272659 0859209401 6488444284 0407916859 8900537637 1163127764 1615386924 1507020820 2977027472 1665334398 0630270583 7049117893 1298272749 9324559554 6340520233 0373997599 6320461488 9408314755 3675360262 9071406675 8743518595 3790747060 0327537761 6769356198 7878180937 7813145173 1616702073 6695808017 5849840525 0495232540 1097455267 5336474841 1047693495 1385758625 4189802064 6520458247 2044302931 7901496889 0775197794 9959382675 0851534876 8166732273 5861168953 5774489505 0416347109 5176950474 0768666008 8353468246 8549415884
• c₃= 1529714 4431639641 3437298441 1101547308 8304694412 2289233289 2592124503 2572072918 6719058935 7533082583 9550200533 9442679346 0473830175 9453670513 4098116403 7447946448 7300141943 2480105985 5341848863 2156017464 6595206890 3740189060 7690167616 5199109876 5309742933 7153779103 4226506993 5259600599 0230110324 3376312593 7570904789 1854963518 1109625110 3685115511 3760778062 5215992774 0903086457 1303056405 9691336883 6566500264 6852672360 6274510558 4679414367 9783143419 3207024011 3906311293 6497495678 7525514060 0106319738 0451009158 8729810342 4123075270 6951669016 4603141205 6769649764 2349218250 2408006732 6140635228 6424676037 8487905119 8476190861 0266207891 2470674837 4149324897 7104147748 2837838602 8307626165 0633296789 8508033143 2840578465 2471473309 2370148321 7360160642 6370309577 5027749803 2476052151 3589475581 0172857484 2313915954 8279037998 8737172278 6607026314 4723792201 5639942631 8417270534 1819974839 9639138544 9605329168 5116208361 5063519613 6957230164 8301011648 5746578347 5832471098 6522639172 4689900467 5456500373 1614671121 0890200538 8361215960 2122960711 3375129305 0267375630 7344606771 8425600171 5325376414 5799781672 6616658515 5991368909 6375220257
• c₄= 14633 5239227321 2436632822 7654769064 9689688263 7060434073 9326591038 2259629089 4521929832 4434125671 1402561341 0890243863 4287699952 2759449808 8912619805 7141319395 7485793985 5300563001 2621505298 9853321870 4943803090 6617752358 3042068348 7733243160 1886134286 0675484755 3271632895 8898680672 0078089326 8999619873 2150904617 1752046437 7701705666 0013405675 5613122659 7621118630 1761352574 5516620768 5574013977 3984404891 9678663297 8251198358 9870961830 2571647726 3166108492 6035577812 5664471829 6310115853 1051349774 1007971373 1463903815 7685223021 4669160366 4104969080 3494921296 6873121656 7495079533 5396103093 7166750176 5452200573 0268727012 8185637638 5478690993 0424254979 7079949676 4722422792 5656609409 9273100365 1245199747 6856690649 6050582350 7524242938 1029187078 5587655295 7229029530 5994804273 2672578144 4624339762 6266483252 2341661509 6947582337 9750970715 9098975787 6016790026 2607337387 7216470726 0645483256 1688209627 1649439816 0628708000 1888030578 0687011863 2508292136 2275579375 1850177779 6745361881 9464982655 0527487647 1310358780 4086095141 2596943877 3842006062 7847420055 2270571282 4718052958 5002583292 9165222557 0217585107 6416901363 7802649428 1683187040 9052629792 5696526907 7861455586 9151586098 9359194394 9316595828 6731560163 0505099165 1225713166 2543683663 6030428519 6589640325 9205576864 3708683605 6696881983 2064169632 2803077941 8785940561 4831121274 1025194779 6681782474 6861000725 2014558365 7654049609 4252315673 1567326403 5632047384 2456699603 5360185195 0492378545 4040160795 5326725375 4347920650 6780765292 6918579454 8146186439 2179435756 2130702771 4038077649 7793899125 4358524364 0531049834 2949069902 9194082029 9042384522 0803726569 9571137277 1627641458 7239085325 9895469623 1106098065 4484927027 9332286224 9671339045 5727169560 6282809921 1064012643 4213837782 6798391089 8331507204 0632186534 4524649812 7530472191 9810645635 5893564675 8273136363 4638997417 3392308319 4732404640 9707956727 7752977389 2626770094 0483899583 2017329874 3864031927 9139979493 6307473065 3905364721 8148972670 2415053417 0044069035 7439696460 6592889900 2366122331 4733837393 2061619059 7141858588 4875847522 0527765009 9144088935 2929431615 8622320113 7407656578 5375728004 4859154854 8059649723 9878371738 1130457847 6411891945 2916472710 4459801516 9041984441 4439897817 5891087652 6352532572 5192062018 2193805609 9908366435 9855092278 1824394102 1914528488 4639669491 7844828171 9653142723 9480137064 8490381986 3544328725 6174409780 3308745744 2861675597 5350512259 7301786437 5443564902 7356616228 6073716556 0204633978 3937487502 4792479462 8708001668 0060641863 9381118317 8797360060 0851450988 9823303933 9581610065 9182343799 4649746508 5800965802 6348925031 1913016023 4317645471 1132804210 5818494167 7287986974 5588620776 4860819493 4672214245 5793645788 9401538083 3006731219 4681079990 1858900558 8241552132 3901597656 8862162342 5942027902 2300270722 1379903357 3596081415 7349299721 3173147254 5856105591 4496668148 8194461647 9920225506 4941532205 8670088559 7168689677 2898029527 1078414979 4117298776 9055818485 5630606655 4776449096 6431728365 3424694464 0990060380 3936526152 8192656665 4467332725 5784076580 4720482606 7633700955 9495478939 9611702729 6403591382 9369166276 7759778027 3568367561 6774664040 1600788361 7107991666 2889167132 0603471099 7838207007 8304753555 4206205449 3743848263 5621537133 5433310187 1887424432 3402885294 1651548784 5377171812 0044622663 6404291987 2978674996 6058832357 2028201622 1192950916 9311004510 1509035026 3780163827 3833915735 4004381363 0147135909 6869756562 2052290597 4790420949 0923937425 6340778642 8413815303 6774326103 2867144425 6840775951 8105019576 9454533667 1460919409 4526453204 9523175250 4726828531 5411998316 0230261636 0167937776 8720430668 9750716051 2807427964 8048020351 2569939301 3787958109 5333457558 2420776598 3760291502 7333456072 3611646369 5210680731 4055155551 6014085135 9887922833 8794714711 0835517283 3360103579 5985064682 7355610775 7953497832 9631638873 5557702731 0991304195 2268304963 3801705252 0168277069 9689784658 3479717490 2476952028 0543801603 7192836085 9446543571 1107841327 7454900764 6023935532 2611161895 6274234014 9245921076 8165042934 5439177898 1076226663 1749732710 8920424369 2114701597 9227992152 3278919753 2224674261 8779059680 1415572008 0212069572 1603753609 3302826349 5192576663 9185541312 1857485619 9196188894 2663517212 8643536011 9094556368 6885636092 3760401125 5482927800 8447035593 8934468881 8607088736 4295878553 4250874703 6255425360 2433938307 2116644594 2415085273 9318626537 7162809368 3252636698 7691607547 0971589021 7232314775 2976891468 7605918900 7625056392 7320048177 9322712023 5865362388 0056128994 8380410758 4558520466 5533140288 4760682394 2230658757 1859427284
• t= 823924 9201034836

Square Check

We prove that (c₁+t·F)²+4·t-4·c₄ is not a perfect square. Here, it is not a perfect square: it is ≡ 3 (mod 63).

Continued Fraction

We approximate c₁/F by a continued fraction u/v such that v is maximal while remaining less than F^1/3 = 985325 0368231061 0478733402 2396561611 8562579592 2700199775 8880942628 3804878754 6081412970 5874867709 5755780649 8764821637 2839151383 7761749504 6426593385 5371697412 0115159317 5897230095 0327861761 8405305598 0423364911 6717055957 7121544389 6443067223 6575096664 9866220997 7407923331 0833267071 0404648015 1095283023 3143599429 2592976174 9724015619 0253315017 3303538608 7149834129 8660002279 5978216559 4222428090 1314851577 8144181898 4509035230 3175140680 7189789312 5188953910 2030834166 1117299199 2789271942 0353905140 6179707958 8575894376 7792207428 7072814172 6748390934 2471900170 5553561682 8891571604 3896502879 7238758601 9224124953 5982865729 2807033777 0079153172 6439645275 4484428263 9565900887 0411914242 0012689421 1804410392 9058757741 0068724467 7604467185 2325047790 2877506453 8565780993 0529191994 6146358278 6327445899 3356114328 3031792073 6047784075 5461584857 5621226138 6071234765 2948395237 3440964066 5290674981 6794938359 2864729646 5364745302 6806873058 2177179575 2605295647 5329788523 1122850491 7803398265 4514078958 1021906747 2471625881 1463433146 1611307596 8051294562 2016343138 0165004099 9578965366 2236285087 7493338176 1400510225 5395966855.

With those constraints, the unique continued fraction is: {0, 1, 2, 1, 1, 4, 1, 562, 1, 1, 7, 1, 4, 1, 25, 1, 1, 1, 6, 6, 1, 32, 1, 2, 1, 1, 8, 1, 3, 1, 42, 1, 7, 1, 1, 1, 11, 5, 3, 11, 1, 10, 2, 4, 2, 2, 1, 1, 8, 27, 2, 16, 3, 7, 3, 2, 1, 1, 3, 3, 1, 3, 3, 1, 3, 3, 9, 1, 1, 1, 10, 1, 4, 7, 2, 156, 1, 14, 1, 1, 2, 743, 1, 1, 1, 3, 2, 1, 9, 1, 14, 1, 1, 3, 86, 10, 1, 10, 1, 1, 2, 1, 2, 4, 1, 1, 4, 3, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 4, 28, 1, 8, 3, 1, 2, 1, 2, 1, 9, 1, 4, 5, 15, 1, 1, 4, 1, 13, 5, 27, 5, 1, 6, 1, 7, 15, 2, 3, 4, 2, 1, 1, 1, 15, 4, 92, 3, 1, 89, 1, 1, 16, 1, 3, 1, 1, 119, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 2, 1, 7, 1, 6, 2, 106, 40, 1, 4, 8, 2, 2, 1, 4, 2, 7, 5, 16, 5, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 11, 2, 1, 4, 3, 4, 2, 2, 10, 1, 4, 7, 13, 4, 9, 1, 3, 17, 4, 1, 1, 146, 1, 3, 1, 11, 5, 1, 5, 1, 1, 3, 1, 28, 32, 1, 1, 23, 1, 1, 40, 12, 64, 5, 1, 1, 2, 29, 1, 1, 1, 3, 1, 1, 2, 1, 1, 33, 1, 8, 1, 1, 7, 1, 2, 1, 5, 3, 6, 3, 9, 2, 1, 8, 1, 1, 1, 1, 1, 3, 15, 9, 1, 2, 56, 1, 6, 13, 2, 1, 2, 1, 1, 5705, 1, 1, 2, 2, 11, 1, 1, 2, 8, 1, 279, 21, 1, 95, 2, 1, 1, 7, 15, 100, 1, 1, 1, 1, 8, 175, 2, 11, 2, 1, 1, 1, 1, 1, 1, 1, 11, 3, 1, 1, 33, 3, 1, 2, 4, 2, 1, 1, 7, 2, 1, 3, 1, 3, 2, 39, 1, 3, 2, 159, 1, 1, 17, 16, 3, 2, 3, 78, 4, 1, 2, 20, 1, 1, 5, 2, 1, 4, 1, 1, 8, 1, 9, 1, 36, 1, 1, 3, 11, 1, 3, 9, 2, 1, 3, 3, 5, 21, 1, 3, 3, 1, 101, 1, 10, 1, 9, 49, 1, 1, 3, 1, 4, 1, 2, 2, 2, 2, 1, 1, 2, 6, 11, 15, 1, 20, 2, 3, 8, 1, 9, 6, 1, 4, 2, 1, 290, 1, 12, 1, 2, 1, 2, 1, 2, 4, 12, 1, 2, 1, 1, 1, 4, 2, 3, 7, 2, 5, 1, 1, 4, 5, 2, 1, 1, 1, 2, 8, 3, 2, 1, 7, 1, 1, 2, 4, 2, 4, 2, 2, 10, 1, 14, 1, 5, 1, 4, 87, 8, 4, 17, 1, 9, 7, 3, 3, 2, 5, 1, 3, 3, 1, 2, 2, 6, 1, 1, 1, 1, 1, 2, 6, 2, 2, 1, 2, 6, 1, 1, 1, 3, 6, 2, 1, 2, 1, 2, 4, 2, 1, 1, 1, 2, 2, 15, 2, 2, 2, 4, 1, 1, 1, 1, 355, 1, 1, 1, 31, 3, 2, 1, 9, 2, 5, 1, 10, 1, 2, 5, 1, 6, 9, 1, 1, 1, 1, 1, 1, 51, 3, 1, 5, 1, 11, 21, 1, 3, 2, 2, 1, 1, 1, 2, 1, 600, 1, 1, 11, 5, 1, 4, 4, 3, 1, 2, 1, 32, 1, 1, 2, 2, 3, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 7, 1, 10, 6, 2, 1, 1, 24, 33, 8, 3, 6, 4, 1, 2, 1, 1, 1, 8, 1, 3, 1, 1, 2, 1, 7, 1, 38, 372, 1, 1, 5, 2, 2, 1, 1, 1, 2, 2, 10, 1, 4, 4, 1, 10, 3, 4, 3, 2, 4, 1, 1, 1, 38, 1, 2, 1, 1, 1, 1, 17, 1, 1, 1, 2, 4, 2, 3, 1, 1, 2, 3, 102, 2, 3, 68, 1, 1, 12, 1, 8, 5, 1, 2, 1, 2, 1, 2, 2, 6, 1, 18, 16, 1, 2, 5, 2, 1, 1, 2, 9, 4, 1, 1, 1, 4, 1, 1, 1, 11, 1, 1, 2, 2, 10, 5, 2, 1, 6, 13, 1, 1, 1, 5, 1, 2, 2, 3, 2, 3, 2, 2, 1, 4, 1, 2, 1, 4, 1, 5, 3, 2, 1, 1, 30, 1, 2, 7, 5, 1, 5, 1, 1, 2, 1, 1, 11, 155, 1, 4, 7, 1, 4, 1, 2, 96, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 6, 2, 3, 5, 3, 1, 1, 2, 7, 1, 15, 1, 1, 14, 1, 32, 1, 4, 3, 1, 364, 2, 6, 1, 15, 1, 1, 7, 1, 13, 1, 1, 7, 1, 1, 2, 4, 1, 1, 5, 1, 14, 1, 2, 1, 1, 4, 2, 12, 5, 6, 11, 5, 1, 3, 1, 1, 1, 3, 3, 27, 1, 4, 1, 5, 1, 1, 2, 1, 1, 4, 3, 1, 7, 5, 1, 98, 46, 1, 2, 6, 2, 3, 1, 7, 1, 9, 1, 1, 3, 1, 2, 5, 1, 28, 3, 1, 1, 1, 4, 5, 4, 8, 1, 78, 2, 2, 1, 2, 11, 1, 6, 2, 2, 1, 2, 1, 3, 1, 1, 32, 2, 2, 5, 2, 3, 4, 2, 2, 1, 1, 6, 1, 1, 15, 1, 3, 2, 140, 2, 2, 1, 1, 5, 9, 1, 4, 11, 3, 2, 2, 3, 12, 1, 1, 2, 2, 7, 1, 6, 24, 1, 1, 7, 16, 1, 4, 1, 5, 1, 3, 110, 1, 3, 1, 4, 1, 7, 8, 3, 3, 4, 2, 1, 1, 1, 2, 1, 5, 1, 2, 1, 10, 1, 3, 5, 1, 2, 18, 1, 8, 1, 6, 1, 3, 12, 1, 2, 2, 1, 1, 6, 2, 1, 2, 3, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1, 10, 13, 1, 1, 5, 1, 1, 4, 8, 3, 9, 7, 33, 3, 2, 10, 1, 34, 2, 1, 9, 1, 1, 1, 2, 12, 2, 1, 2, 1, 1, 2, 4, 1, 12, 2, 1, 2, 4, 2, 7, 8, 4, 2, 2, 1, 2, 1, 1, 3, 157, 2, 15, 1, 1, 43, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 6, 1, 4, 3, 1, 1, 1, 4, 1, 2, 1, 1, 4, 1, 73, 3, 2, 2, 1, 1, 4, 6, 10, 4, 2, 2, 520, 1, 2, 1, 3, 19, 1, 6, 1, 1, 1, 3, 14, 47, 1, 1, 1, 1, 2, 23, 1, 1, 5, 1, 1, 7, 1, 1, 1, 1, 1, 4, 1, 2, 2, 2, 2, 2, 2, 1, 3, 4, 1, 2, 4, 1, 5, 2, 1, 5, 13, 1, 2, 4, 161, 2, 2, 1, 14, 4, 1, 1, 2, 16, 1, 20, 2, 2, 2, 1, 6, 1, 31, 2, 1, 9, 1, 14, 50, 1, 2, 1, 1, 191, 4, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 3, 80, 1, 1, 1, 2, 4, 2, 6, 1, 1, 17, 1, 2, 2, 1, 2, 9, 1, 4, 4, 2, 397, 1, 1, 5, 5, 2, 5, 76, 1, 1, 3158, 1, 4, 5, 2, 1, 19, 1, 4, 1, 2, 1, 1, 3, 3, 2, 3, 1, 4, 4, 4, 5, 2, 7, 1, 20, 52, 1, 1, 1, 6, 1557, 1, 1, 5, 1, 15, 18, 3, 1, 1, 9, 1, 1, 1, 18, 9, 1, 1, 1, 1, 1, 1, 32, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 5, 5, 1, 4, 6, 1, 1, 1, 1, 1, 1, 7, 2, 16, 1, 1, 38, 1, 5, 1, 14, 1, 7, 6, 1, 10, 1, 1, 3, 1, 1, 1, 3, 2, 7, 2, 6, 1, 1, 4, 1, 6, 1, 3, 78, 1, 1, 1, 55, 1, 1, 3, 1, 2, 12, 17, 1, 16, 2, 2, 1, 98, 3, 1, 8, 1, 4, 1, 10, 2, 367, 1, 3, 2, 1, 5, 1, 2, 5, 1, 16, 1, 31, 1, 1, 1, 2, 1, 2, 1, 2, 10, 2, 13, 1, 5, 1, 3, 3, 1, 4, 5, 1, 1, 2, 3, 8, 1, 2, 1, 5, 3, 2, 1, 1, 1, 7, 5, 13, 2, 1, 4, 1, 43, 1, 3, 3, 2, 2, 3, 4, 5, 1, 7, 1, 1, 86, 1, 1, 32, 2, 8, 1, 1, 2, 1, 1, 7, 5, 13, 8, 1, 34, 2, 2, 1, 148, 1, 1, 2, 1, 1, 1, 9, 3, 2, 1, 1, 5, 1, 2, 1, 3, 5, 4, 1, 7, 6, 1, 6, 19, 1, 1, 1, 1, 8, 5, 1, 3, 8, 1, 5, 1, 7, 1, 1, 3, 2, 4, 4, 1, 1, 4, 23, 1, 1, 3, 2, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 75, 3, 1, 1, 8, 8, 7, 4, 7, 13, 2, 8, 1, 34, 65, 1, 3, 1, 1, 1, 1, 8, 1, 1, 38, 1, 2, 3, 1, 23, 1, 4, 8, 8, 1, 2, 1, 2, 1, 1, 6, 3, 6, 2, 1, 6, 12, 6, 5, 1, 1, 6, 4, 9, 2, 4, 13, 4, 10, 6, 3, 3, 17, 1, 3, 1, 6, 1, 1, 24, 1, 4, 7, 1, 3, 9, 3, 1, 1, 3, 23, 1, 2, 6, 1, 2, 4, 1, 2, 5, 1, 2, 16, 7, 1, 4, 1, 17, 9, 3, 2, 2, 1, 1, 6, 4, 6, 3, 1, 5, 4, 7, 1, 2, 13, 1, 2, 2, 3, 1, 1, 1, 1, 25, 3, 3, 13, 1, 1, 1, 14, 22, 1, 31, 3, 1, 7, 2, 3, 29, 3, 19, 5, 1, 3, 2, 3, 1, 1, 1, 1, 1, 2, 4, 7, 5, 2, 1, 1, 1, 1, 27, 2, 1, 5, 1, 6, 1, 32, 1, 2, 15, 2, 2, 17, 1, 1, 2, 1, 7, 5, 4, 1, 1, 10, 2, 20, 1, 1, 1, 1, 17, 7, 1, 1, 1, 13, 748, 1, 1, 4, 5, 1, 1, 6, 1, 2, 7, 24, 1, 1, 2, 8, 2, 3, 14, 1, 1, 3, 1, 9, 48, 3, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 2, 4, 1, 1, 921, 1, 2, 3, 1, 18, 1, 1, 54, 1, 1, 1, 9, 17, 19, 1, 1, 2, 1, 9, 2, 4, 31, 1, 1, 3, 9, 1, 2, 4, 2, 2, 2, 1, 3, 26, 1, 1, 2, 1, 96, 2, 2, 7, 1, 1, 23, 2, 11, 1, 9, 1, 4, 2, 1, 3, 1, 6, 3, 2, 1, 4, 1, 6, 1, 2, 1, 2, 3, 1, 1, 7, 1, 1, 11, 1, 3, 4, 1, 1, 1, 4, 1, 5, 3, 17, 1, 2, 2, 7, 1, 10, 3, 1, 1, 4, 1, 4, 3, 1, 1, 3, 2, 1, 8, 1, 20, 1, 11, 2, 1, 1, 2, 1, 4, 1, 2, 3, 5, 2, 2, 1, 5, 62, 1, 1, 5, 1, 2, 1, 27, 3, 10, 4, 4, 2, 1, 1, 2, 3, 1, 2, 1, 2, 5, 2, 1, 1, 9, 5, 6, 2, 1, 4, 4, 1, 2, 1, 5, 2, 111, 4, 39, 1, 1, 1, 2, 20, 27, 8, 1, 1, 1, 1, 12, 93, 5, 43, 1, 2, 11, 1, 13, 3, 3, 2, 2, 21, 1, 3, 2, 1, 7, 4, 1, 2, 1, 3, 1, 3, 7, 1, 8, 1, 1, 6, 1, 1, 39, 5, 6, 3, 1, 1, 6, 2, 1, 1, 2, 1, 2, 1, 1, 8, 1, 3, 2, 3, 14, 1, 7, 137, 1, 4, 13, 1, 1, 2, 1, 28, 1, 4, 2, 1, 1, 5, 2, 4, 6, 20, 1, 1, 1, 4, 1, 4, 9, 1, 1, 10, 16, 1, 1, 3, 3, 3, 4, 2, 16, 10, 1, 1, 1, 10, 4, 2, 59, 1, 116, 3, 3, 6, 2, 6, 1, 2, 8, 1, 1, 1, 6, 1, 9, 125, 4, 5, 10, 1, 2, 1, 3, 3, 4, 1, 1, 5, 3, 2, 2, 5, 1, 55, 10, 1, 3, 1, 2, 57, 2, 1, 1, 6}, giving these values for u and v:

• u= 2396 1052928172 9542663695 9752312477 1611787494 5098170462 3818455802 6347569679 0208319316 2795660634 8609896370 0713141516 3232583207 8678293565 8923377998 3730903012 6449038335 7940097585 5076776369 8510531673 6082246982 3162148716 4281186335 7386806569 9603625679 5380214538 3561867276 9025072730 9688698636 9928590344 4656090471 7947418264 5558575466 6637740100 1834370946 8472440063 5803157205 1051734400 8456386401 8250439014 5786257922 7826924796 4096009727 5251407275 2930161335 6600772520 5115823792 9816273900 8661876742 0509782414 9426940940 6726218681 9569555588 2999257591 2076864971 9273981623 6882501329 2323868440 4221291111 7814379503 1170028857 1422770484 3197855092 5986509960 5842763157 0076809533 3095767418 6501667806 7815969704 8046673534 1755728761 3366549475 6349808212 5788156364 0512289174 6287585607 1782535704 0530399168 3426673590 3078424962 1420552902 1056941183 0495668125 4861844086 8187915686 0159146950 0749242143 7137930019 7372030642 1472615782 9342194649 1811168538 5520652755 9708950556 1358559694 0896160112 5634611078 4406081466 1898010065 8899956722 5775556642 3274192714 8475072307 0027064661 2489274989 3346774138 9664367607 0235130877 5650894061
• v= 3337 4269470048 3469834534 4234914865 7211805916 0444483496 1303785307 7679509453 0528243878 4824720654 8182366716 8198645734 8303989612 8357931812 6043114720 2821114700 2789194310 4553250528 3273758807 9198061585 6360775446 4334774523 0423681772 3626814698 4876775131 2526980795 7203552330 3853535666 6365081260 9062576762 8897662868 5127571089 3219325733 0965783244 1488519925 5387084716 5930067735 8592182714 5010455910 7770454035 4527975087 4861589953 8802127290 8367690386 6489935534 8720572818 9393825330 5217386407 8906060121 3066764153 6536312169 7877410629 6046241994 3947369831 6774266411 8612551585 0571088731 7945541199 7731279017 5486205187 9182796655 1918401007 9428388038 5288376955 6446843055 3307415565 0133610032 7191883483 3738801802 1727431001 4443548904 1790118271 8119422825 2949499874 0406751829 0004065651 9797597208 3816777768 4516271679 3471218432 0997592162 8411381284 7936442619 4073468730 4935086925 8599337411 2030297492 4578867905 5941631886 2645073518 2540055875 4465567535 6580477237 7090130364 7173369424 5139532617 7411275639 7146641647 1392106303 3496697495 4300826578 7086231385 7144125528 4485554595 7996539239 0482139336 8831467516 6871949109 7191707544

We also need to calculate d = floor(c₄·v/F + 0.5) = 5105310203 8379095492 8514111750 6433778506 6021227587 2535894454 2500362199 2742490467 8103053542 9351636168 8376678422 8194991617 1950356751 8186964781 1923362337 8231235124 5668349241 3105477968 9192956320 5545569701 3722520165 1213154677 8177617301 2497777102 8898700205 4464326849 6732153792 5888013095 5419640772 4132713657 2423745986 0661754329 3865208741 6661641566 4473539901 9269347575 4152555150 6836361972 5616813652 4496063900 7235910406 1835583468 1544475049 7449760565 6171237936 7824934022 9417980852 0584382572 1138597229 0468101678 4442664342 7665000114 1240075608 2324221791 5360634091 6260551142 5189714306 4927204813 7704131609 8835641898 8208167761 2353729562 1295566328 6184436455 3187570156 9966719921 0292299824 5554252758 2705860520 2051749018 6245540203 6481183023 1510117582 4875988384 0857285554 9547489992 5723804550 8896739525 0426503439 1516295593 0629258876 6195565687 3632178896 7721795070 3835484273 5247170431 2231184961 7298357569 4147696548 2918099766 7181338808 2478927150 3511668376 3709503603 6932764609 3752186645 2416831926 7171064464 8211269524 2332482607 9328430915 2069148100 2893290971 2049138738 0051962854 6899331573 1266354655 1462267985 2470720167 7421901520 3645497504 2781949567 5588896448 7978780437 1131748352 6497315925 9698399810 7568173835 2655576984 7922109094 7894415166 8322631612 8003228067 5280198322 1014968274 7099733809 3682390343 5911518729 3074409612 7265899610 9471169087 2986585341 0838735686 9669878227 9255868674 6640499159 4549658523 8394206128 5625823153 8494188239 9351411311 3025421713 3409225813 5063567099 8232938767 8260830694 8952539855 4563733434 2712118137 7261479716 6369991663 2844591104 8900962470 4893205501 0138481813 2024361388 4836160694 4086702978 0740289365 0468224958 6903232831 9796054595 6206287116 4702242950 4306118419 9986709393 0310028897 0633542940 5559164181 9722198048 7172142176 9092986451 2002162545 3744707568 2361545348 0249197264 0521581780 7926220635 9825714541 2116298424 6486980501 9310395913 5468383849 7344987156 9939447573 3394440929 1646589592 6564616721 8361153603 3865334106 9273407052 5860339691 3774669590 4774171951 4800666246 3970227500 2922321700 2151469802 2819496344 3732333884 4036279760 9867786514 7274560912 9252702124 5533588735 8330621519 9936499808 6956021077 1921795167 3766464895 6793125064 8729194178 1506281271 0251087547 3342367046 5707687589 3932928606 8330522125

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= +3337 4269470048 3469834534 4234914865 7211805916 0444483496 1303785307 7679509453 0528243878 4824720654 8182366716 8198645734 8303989612 8357931812 6043114720 2821114700 2789194310 4553250528 3273758807 9198061585 6360775446 4334774523 0423681772 3626814698 4876775131 2526980795 7203552330 3853535666 6365081260 9062576762 8897662868 5127571089 3219325733 0965783244 1488519925 5387084716 5930067735 8592182714 5010455910 7770454035 4527975087 4861589953 8802127290 8367690386 6489935534 8720572818 9393825330 5217386407 8906060121 3066764153 6536312169 7877410629 6046241994 3947369831 6774266411 8612551585 0571088731 7945541199 7731279017 5486205187 9182796655 1918401007 9428388038 5288376955 6446843055 3307415565 0133610032 7191883483 3738801802 1727431001 4443548904 1790118271 8119422825 2949499874 0406751829 0004065651 9797597208 3816777768 4516271679 3471218432 0997592162 8411381284 7936442619 4073468730 4935086925 8599337411 2030297492 4578867905 5941631886 2645073518 2540055875 4465567535 6580477237 7090130364 7173369424 5139532617 7411275639 7146641647 1392106303 3496697495 4300826578 7086231385 7144125528 4485554595 7996539239 0482139336 8831467516 6871949109 7191707544
• z₂= -91 1715625261 2161740995 9996326659 4367624102 8247761885 3051670599 3250721293 2578383051 0382820346 9753383109 9216267019 2569245401 5432059635 3784365255 9186139762 5699872275 6118451318 6056000577 4012448138 1315686977 4701116475 0229108050 9781228107 9882207936 7491664638 7663975779 3209316811 5844125962 9332275697 0429051776 5275330868 7563902697 9080129201 1090850053 5042352905 3381492693 2705897703 4103380729 6976394536 3816050693 0616590259 6318377766 8131633532 4842852720 4581817031 5543379975 3044651715 9095050095 2001973472 9733285326 8899000828 5554835532 7076085636 0293604232 6593293012 4206899090 9876977142 5339835552 1368335913 4367650789 4248266734 2513514109 4609367946 6857640357 2311919243 3657312654 8486068937 6187655710 2193246504 5805214199 7392324050 1147324638 3867390440 5144859781 2020110984 2980578952 5965105817 7931241449 1144634603 5301599535 5030551633 0864961562 9210064710 7351298712 0247275169 5501933548 4013807962 1872764645 7350533626 4427227787 0973315435 0530345301 4467726736 3973600999 1912173948 3778357551 2458579254 1491793629 3608445098 5743565124 2899125815 0066295433 5189091993 2974528508 0080870405 8487376366 3275964702 5435783844 0752821684 0515039238 5201875336 4230152602 5709555843 9881503762 1798655872 8017968834 1024750058 8520104995 3596115063 1274435314 6252201589 9639384566 9277277776 6653050906 0647059751 6426181511 6497239487 8711843337 5482549665 9784430536 5614478490 5217223725 2918242579 2362885862 5567986816 6675998009 0007705836 8488291827 0394051149 7242072927 4066122368 1668906787 9188695490 7437290803 8935515443 7353768057 4852690871 4541438540 7773528324 4695111676 2291600779 4558908527 9166205577 9990011188 9889579392 5303617816 9840467182 9120831555 5691266809 7310640120 6251786301 8712109313 6612051648 9994502925 1374824892 8139595151 5648004939 9578768498 4210565227 7799699269 2304690886 4756585607 5363130856 6763370661 9128351610 6717522471 2982761041 6114126171 2428370588 9314414536 0839376634 4905728349 1570110266 3942283089 7362937852 3298646402 3141146987 5461474355 4363096598 8116316694 3139370734 3845020317 7399561068 2012031582 3545933489 1403627763 2847518491 1911296289 4195187882 9229764985 5517347269 9890561256 2923516826 1846128804 7296449248 3940962129 9165296139 7245224456 0754688993 1044073040 5751793002 1885859667 9287557329 0725458452 3627523343 0578531270 6929901456
• z₃= +38306275 4265557242 1993832188 8605356012 8142693220 9475341069 1587678909 9949962257 5572901463 6121884925 4935409393 7922447762 0627616977 4066917449 4514724622 6584473477 9540889331 9588191988 7921058654 1761519052 7854541983 7834101472 3504527753 6757984235 4237787612 9455864711 5965474421 7554982032 5694277068 9079186757 4976439071 8499076124 2484633308 6990608913 3974292122 8218055462 0997078875 8999909145 2194778885 2875583531 7000801706 1541927764 7799353272 2762296492 0367174944 1516757818 5180374852 2011832189 6676609524 3555960241 0451272223 2967484706 6828781011 6044691083 2954393646 5060406496 7968766638 7871384252 9881257422 7535979763 3434890222 6513303947 5193701221 0866173476 4205241273 4308231386 7383500744 8308786223 4801812679 0921900090 2098861963 0403625314 4060246289 8477864356 3269525143 1917514817 2624559300 5394987912 3209340555 3485716176 2382482938 2030093861 9208202070 3570582929 0721112358 5816686177 1174224510 9251937519 2031421077 7349568564 1589514208 5516736197 8991522543 7417268553 3846577383 4614318351 4977122465 2981897732 2414294451 0769664886 9213105365 4278754198 8221925451 4797199322 1005604415 1813382458 2771112514 0541706991 4951290472 4888067756 7510932863 3217075131 0617197172 2118090342 9759083106 5745073951 8879291384 6816397465 9381939223 8723227434 1721171427 1946956788 9297037679 9947972552 6360995886 1553111819 8762721410 5509494753 8882788527 5252076939 5039452010 8505799846 7150415996 3618608555 7175906312 1781102205 4283241000 5353789830 5884221799 7777451356 4893862465 0121844300 2957407301 0429995007 0735425961 7058413336 9699777793 8870865711 4123731114 4835019202 6832341348 7565648000 2713149828 9283318758 0421350365 2028141350 0564637896 1061245345 4770357091 4569606168 2588125121 2497061104 5134638642 3133977240 9765593248 3208278076 4134919827 6351161589 6915022027 4771344836 5753486515 9856414481 2374345770 8003537424 1921117318 3704691683 5858344692 8250546237 3834874585 7678790845 5439459167 2012283414 8076835227 9791295698 2126058682 7647658581 2738925537 5774930624 2574487102 1483659362 8232666847 9715977409 4915942003 6082562136 2657130426 3832571993 0709311818 0693529393 3533684682 9654648525 7581399267 0168658002 7948131254 3803730210 7570819437 4153997306 3450299621 8375056338 1912418541 0571408842 5878635318 8905144024 2208020969 4073378319 6637278314 6512427846 3184729457 6809569152 8929518752 4581070700 1281919201 9060308342 6860678033 3369968684 3559266997 4458481292 0601920295 6963212276 8011312700 3824647667 1304419507 6065412883 2283641858 3672094455 6460109783 3913362469 6373495739 8681851315 0835631907 6602095752 5365326787 7538930022 7183572557 5322019867 2274618497 3180733830 5903713096 6893216142 1347270852 6330912653 6583769938 8294119005 2537415080 0844599155 5772878847 1846737104 3787133597 3568711858 6152725099 5441848430 6193503930 9647987157 5000457097 2246381344 5136301165 5614751532 1542788845 7940303026 6187854588 0336040368 0195523907 4542189115 1661227330 1629730308 1527012880 8352536332 0712456833 9047451417 6697623787 3218381442 5097514635 3088457155 2551365150 7839109982 0703797296 1027355690 3283829545 3712241638 2779598837 0681405862 1675322257 7690012113 7070181539 7626342298 6586923092 6648218884 7667391575 4250669388 8101687038 8513984588 2001462564 0196199058 0973600718 9170160507 2017066640 1195374882 4483024754 3952637230 2074036956 8337751303 1406760321 3864825233 1340617287 7360938475 2887445206 4142000700 8357444096 2578216022 5861815121 3396383544 6877347065 1128046133 9991781484 0389293882 0498627002 4963699557
• z₄= - 5105310203 8379095492 8514111750 6433778506 6021227587 2535894454 2500362199 2742490467 8103053542 9351636168 8376678422 8194991617 1950356751 8186964781 1923362337 8231235124 5668349241 3105477968 9192956320 5545569701 3722520165 1213154677 8177617301 2497777102 8898700205 4464326849 6732153792 5888013095 5419640772 4132713657 2423745986 0661754329 3865208741 6661641566 4473539901 9269347575 4152555150 6836361972 5616813652 4496063900 7235910406 1835583468 1544475049 7449760565 6171237936 7824934022 9417980852 0584382572 1138597229 0468101678 4442664342 7665000114 1240075608 2324221791 5360634091 6260551142 5189714306 4927204813 7704131609 8835641898 8208167761 2353729562 1295566328 6184436455 3187570156 9966719921 0292299824 5554252758 2705860520 2051749018 6245540203 6481183023 1510117582 4875988384 0857285554 9547489992 5723804550 8896739525 0426503439 1516295593 0629258876 6195565687 3632178896 7721795070 3835484273 5247170431 2231184961 7298357569 4147696548 2918099766 7181338808 2478927150 3511668376 3709503603 6932764609 3752186645 2416831926 7171064464 8211269524 2332482607 9328430915 2069148100 2893290971 2049138738 0051962854 6899331573 1266354655 1462267985 2470720167 7421901520 3645497504 2781949567 5588896448 7978780437 1131748352 6497315925 9698399810 7568173835 2655576984 7922109094 7894415166 8322631612 8003228067 5280198322 1014968274 7099733809 3682390343 5911518729 3074409612 7265899610 9471169087 2986585341 0838735686 9669878227 9255868674 6640499159 4549658523 8394206128 5625823153 8494188239 9351411311 3025421713 3409225813 5063567099 8232938767 8260830694 8952539855 4563733434 2712118137 7261479716 6369991663 2844591104 8900962470 4893205501 0138481813 2024361388 4836160694 4086702978 0740289365 0468224958 6903232831 9796054595 6206287116 4702242950 4306118419 9986709393 0310028897 0633542940 5559164181 9722198048 7172142176 9092986451 2002162545 3744707568 2361545348 0249197264 0521581780 7926220635 9825714541 2116298424 6486980501 9310395913 5468383849 7344987156 9939447573 3394440929 1646589592 6564616721 8361153603 3865334106 9273407052 5860339691 3774669590 4774171951 4800666246 3970227500 2922321700 2151469802 2819496344 3732333884 4036279760 9867786514 7274560912 9252702124 5533588735 8330621519 9936499808 6956021077 1921795167 3766464895 6793125064 8729194178 1506281271 0251087547 3342367046 5707687589 3932928606 8330522125

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:

• 0+ε∈(0,1)
  (Root is not an integer)
• +420804 1656150157 1201751205 1642196038 3580382394 1163701635 6299735529 4161498254 2096807262 6195070997 5554720493 5437479153 0102274158 6820003208 9857010182 8410428921 1096116484 6242686086 4609529799 2740249490 6617939159 9374736958 3676652980 2388200877 8922394717 3000840228 5666698847 3944081161 5983864344 8144947920 1486603273 0129730271 9937935534 3922630042 4722674217 7562564193 5957127229 3560602510 0052826877 2603323041 0664881321 3872137104 5352159305 1223763620 8375993229 8399427407 6764331769 4174596039 3578271597 2058243199 7810774797 1073628642 0932887941 0589313481 6117298030 8129571191 9296290513 6243979379 3868439149 1459373509 1544286854 0527159591 4110660810 9657469541 1480763317 4719082888 0840575866 0462217056 5053192027 4933710821 6651731042 4214348039 7619373729 3778433705 2799838876 8566847525 0872328683 9518235108 0173396970 0361087406 5309619498 4536209403 0004788402 1440730868 6561935384 0593659328 2330181602 1533001286 9449623419 5326945360 5098499647 4951359592 4303076294 3998114242 6026250160 5795488909 0088387368 7941458129 9511145053 3538694693 7581978999 5525516162 0673203337 6180203721 1844080429 6399528734 9139776078 7739919683 9486165076+ε∈(0,1)
  (Root is not an integer)
• +272758396 3797131859 1550443535 0510530132 4444242902 5190363909 3144557236 3350068890 7089130023 8160634897 8639714115 3170595017 3043105597 7341260242 6864715572 7256727320 5732146589 5865048683 7847551855 7788533622 0498703456 4082535956 2308676310 7747822304 8801504763 7014458387 6086286226 1188170049 5486477884 7932926487 9797313317 6868668942 2248680565 1117716217 9678598305 1744916792 3738654843 7504191433 9433143991 4117160782 0845573104 7634233439 6657028398 6909835849 6995625841 0652732450 6821047011 2498145333 7001526978 7692996400 6978346798 6536550157 1666619373 7011781811 9875487587 4717969515 4472017838 8292226191 6864281784 6332524023 4218849097 6451437551 7258750308 6241950514 9237919809 1661858933 5275639075 3150972311 1198783580 9770219827 7145953572 5121691109 7951793755 6871687952 4542385178 0325625030 2959062821 3805558603 3910373568 9547712795 9804161214 4698021669 6264141571 5703937794 0765059523 6778216597 2048493620 3012202655 4752345719 9073946046 6561304791 0929105176 6931539558 2276031469 6508490977 1183794791 1595345669 2180489428 0036431558 6106387406 9094619557 8344212406 9486998181 9510865465 8346112481 6650074790 0137571796 1103372586 4829332057+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.