Primality Certificate for (978^3061-1)/977

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 30.184173% factorization of N-1:

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

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

We set R = (N-1)/F. Note that GCD(F,R)=1 and Log(F)/Log(N) = 30.184173%

Finding a Witness to Primality

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

Express N in base F

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

• c₁= 5 5104469803 6866112432 5096524136 2886276876 6336674760 6672111304 9472451523 5819721960 6238087285 2671458281 0812530547 5831566207 4380278481 2814886971 1923890739 0338671621 9425788604 0977993126 9442577436 5536892071 9879618852 6713632644 6583736153 0920052532 8024614068 9487963126 3362094019 5139901997 1382631684 5810690774 0300370869 0110871295 6687024092 9963289192 9246635596 5308098072 9334430298 5383853786 6935757516 8217741902 7986310763 7360802514 7200412504 4032267717 9743454327 5195018353 9925720912 4412723304 3026553613 1809183436 4879500270 7654094715 4918303603 5078483943 6447694217 5492601753 1213655207 2409477243 4398030103 8483835476 0005968186 2239464199 1610106202 9540441396 2591046325 4450019449 5387652577 0353878840 2190594336 4632924925 0847166311 6902159132 6237899887 2210587734 6845745356 3533487238 9091980042 0705461889 1085730887 9102438474 5299296887 8322425339 4968942979 4000683096 0130475969 1242392352 6943743167 0068973399 1368482972 4558024202 8819355295 9810203142 8286241965 0383553439 6692818735 1839704780 5037767040 1040846390 0094759899 6707988383 3163878083 3849348018 2662680272 5860914977 3426444442 3030139449 1803082977 8139422981 8466662564 0319795022 3778254066 3685940231 2018294731 3288893490 7149802914 5756259751 7932385896 3233407882 8307338805 3331799494 9108589792 3564907515 3644582500 9481113776 6863879466 2976382680 8365921033 9030431326 1310350415 9508459089 4268464415 5203317401 9642264089 8322612880 0565979772 5784154076 8162424977 5426244440 9459976965 3844686144 8812979069 1504278127 7246094354 6482723824 6610107553 1799608276 4196659902 7395108386 3909229572 4843676749 1276134763 6136734441 2055635630 7097247491 5239380033 7324855490 9490119439 0527188696 9370985975 9291832465 1778464123 7225046233 6200862677 9778887987 4020712010 9687524078 9875635799 7515528177 6644091540 3983896217 2179056576 7610758914 1490410853 2758401474 6189541357 0966709738 3815945196 6202703239 4601838326 3672007135 5220592068 0533280253 3043634947 2478090569 3555230053 4592199038 5682337445 2294803745 2851565494 6582259822 2188037289 9874008296 9964371162 9914048779 4071648182 9223672426 1397457090 4732436378 3611573250 0854228172 2051719970 4488383961 8129224460 7835070786 0154183766 6332314574 2125438885 0924904193 9304788094 6327926662 3308172493 5726705591 8122381817 6344020390 3206280148 2840869079 4018970018 4152505934 5518031084 7441853370 7271148561 7150818210 6800006363 9212879523 0518123904 3543327407 2044020777 4769768456 4326248644 8536588503 7075822797 9497192651 4933200292 8577694923 8383197371 9270223460 9578746723 7614713393 8181643231 1591378897 3015833371 7166434550 4933406929 8927206158 4896076162 9856970237 0103887188 4276167488 1885492491 0611614939 6808352390 2249518506 5387464454 1048274822 5908029823 6018555343 0793166825 6158844193 4300387319 6434452407 6547806523 9007704552 9052301934 6511299951 7698799634 2138007234 8356119402 7804853003 2837707812 6037318572 5111704611 0411877462 8166668993 1188856052 3015177699 2247169097
• c₂= 72 7564087129 9770261637 1245099781 2880687642 1810101886 6441209351 7921876118 9083390933 8307934418 2484630296 3544581529 7024215547 2078800240 7782267535 5533488601 6117092007 8467295909 7887173131 3738657018 5732111263 0902048727 2468326432 7605792483 6481029463 0689704189 0271063151 3602702084 2269629867 0287483443 7961309478 9186635727 3773907132 2517456255 3989981563 3261065845 3160748704 0561656492 4413055733 4250015563 2571551927 6298214540 2668041390 3965456088 4018474966 7385834331 0789210217 7728037240 9683787186 4341768254 6186108543 9830616121 0912363274 1874784540 5453325298 6574115473 2444551433 1755326206 0858199856 4798051254 3861733143 5615036339 2964601535 2852262283 6033699938 5269765991 4246565697 7332795089 7305798200 9553372511 5677609116 1368035974 3020009304 4942559740 5760556321 2879980396 4150366877 6132258559 7371626708 9338279447 2779564982 8082114160 1752797957 5287522433 8769373389 7912339870 4901774381 0539054143 1335055887 1643828043 1591061593 9449687604 6350853064 2626785739 7408708021 1162321906 2133925409 4075456074 4788845980 9565295167 4191357681 3681088409 6955424137 1645941033 0011751761 7637305750 0112967578 6935493570 5314170802 1469522915 8300629024 4116771122 3251647337 6636360680 2028892266 8575435477 7000486400 3451689397 5847033983 4189920236 3527298804 5548973118 0287420826 3121587189 0367861687 2702110595 2919378114 7676504416 6377381281 7513648705 1925571100 2682200305 7031787301 8522984257 1246912215 3820315777 1352974219 5587405755 4725096999 1311738715 7806349210 0136482910 9017351132 6457849247 1000606982 2165417834 9505302696 9658316747 7246605832 9790108203 1388230359 0693807623 7829939954 4006972196 3833003648 5642503595 5932894478 1361189498 9157226018 1423193437 8647865434 7044540328 1981386279 9613684452 6959478206 8523159232 4228148857 1688955309 1462414516 7108851494 0606768338 8993460023 4795567972 3579500606 7647617296 5679608289 8364088603 7260594950 8708644977 5251281287 6399932257 9233774252 9615931233 5188584155 8533945007 7948295628 1426377340 0126755985 3997872642 4674636047 3852978544 6906807897 9547887554 2488673664 1026975207 1923088547 1574028811 5364758750 5198771093 9304505841 9076981665 4334918760 8274651290 8546697474 5832869682 7656987335 9205269986 8517426158 5963125673 2097182173 9671843708 7171603276 1849449557 2140367566 4233446263 3519702540 0645356234 1060811338 7287755992 3123358880 3885993823 2589729982 5501581614 0257708793 2777412481 2164808882 1516338375 1596011857 0127195623 7042677720 5823015585 6149687521 5587555009 8724849231 5164534866 3615444412 1052896485 9521184832 8515265682 1480761936 8723562733 2527270768 3015932986 7891027567 9576406569 9415742862 3006179888 7451636395 6494740064 8367823256 4794034465 4899603508 0515895274 7967220372 8794839639 4703724990 7910332343 6158871763 6917981234 9055935147 7708583780 1165195072 8361894286 7453350938 4209638702 5204288313 5151324078 8210421898 5002337999 5336485183 5309405037 5344537852 8742028589 6915823193 0812621371 5846611250 0219159404
• c₃= 30611 7831756637 2762270431 0222282811 1728731613 0663416836 6267698926 2009489840 7490027882 8369846413 2982270826 2599580620 1616248737 4139462645 7160329343 8246903384 0814664673 4618555540 5210993700 5290211143 6656741544 9508342848 8597314779 3787420785 3654006172 5631019756 7213562695 6784410997 1229379328 0526929298 9226121567 5195262965 2348421848 6540319868 6197946748 3705304091 9217943095 7588337678 6676845412 6749897531 9873989360 2619283428 9422889112 4636911518 2105763930 2107813130 5775547931 2699049311 5339631234 2310419457 6056509515 8096903532 1891324149 2873457575 0576730412 9015764145 0567313763 9089343458 9087229520 2347278950 8252295185 3376218024 3327584719 8423675979 3394082207 4519829614 2877400481 8923574411 9995265544 5827693144 7126855044 6627230754 1605003802 4874443711 6034605022 9330609451 5331729039 2209441805 2813030617 4469581270 2866637609 7572096832 9864695474 5091327961 6659187559 1572266485 8516775598
• c₄= 2949210 2371845848 8543889618 9100017231 9708703275 1069383215 6251439179 1129106562 6008095626 8346484451 7590082169 3763044948 8190232091 7605184806 0249758810 8358357761 3434920036 2497006412 8270740016 2569029559 1761485607 6910725969 4130582606 3504690314 9558736726 5109609471 1334847619 4202116464 3315539481 4278356998 5914141148 2817058414 7290233892 4351557896 9822649505 3006004264 2475048516 4446258136 0020161301 6566148408 6451202026 8952591084 2059137254 8871982870 5271299807 3573080201 5274803728 6053332074 9188091154 2257763978 8011566810 0425734291 1083862976 9410739435 3463930584 1919433687 4145237924 0013739115 2195386571 5156486180 8695155859 5095978735 7615764939 9111836757 9643859608 5301342002 6011620474 9631828046 3994953948 7375233895 0257090028 4706637556 9187462962 8335111845 5075027910 8130218560 8113398758 4762945819 9256316590 2032477217 5119036165 0740529794 3162244960 9577452560 6759804311 7060107012 4495572445 5860808620 6635516950 1651133915 1588618838 9186513608 3739814271 5477641809 0975399506 4029339610 6162184382 4256216121 0211156829 2504597161 6684140005 5278249135 3867806435 9171413261 0448946656 2730915161 6705775454 5264534565 0563193976 9640634981 8561895567 5970080618 1556889339 7521664826 1437167106 9251615818 3222689637 9153264713 9541377029 0616336951 8081778647 4489787148 5552757596 3545375432 6740333234 6262980774 7389106217 8637802083 2536047895 6794095570 8861026072 0811059355 1196457444 2547236245 8045725718 1612863114 3368422377 7296577384 9975940731 7140242540 0904382106 0691457678 6365527637 9466676130 5513870605 7343847462 1107957104 3644813196 3199672844 7768514613 2828831666 1322647446 3784351331 9143613754 7697249627 8833979230 2713167421 9349927414 7955875865 0116437585 0584921196 8647216771 6045860446 8257119719 1136411909 5685399657 9526409288 0423922245 1821229748 0518915572 2603686077 8799565317 2183161772 8137140404 1723888837 3221455748 6391957359 7004534414 9044927228 3066360234 0875986488 5361341719 6842194945 0675350695 6015025532 6037373298 5867114044 3036790484 3360343161 8005329323 0442700021 5291336906 3084534852 6594676626 0283696361 5782625927 1941662804 9912766575 3771422679 5962993990 7528200667 4124966655 5131984189 2863416563 9530061223 2421247353 5700364840 9577713753 0041726598 5735329032 2125180314 9043888748 1371450857 5765232195 9701158381 4214666317 1986467454 2215370243 9601026562 2996238222 1367898894 8806878363 1680446689 3430336811 5207020545 6616067750 3548261860 3365452663 1389303946 8980376063 9518672010 3991939567 7510068373 6004976933 8803219369 4017177781 0405044621 3209136030 9411893155 5156047398 5938030164 6592875843 3194762192 0536058958 0825535496 2632267398 7658409350 5558768971 3674255921 0234508274 6913127120 7704648071 2470819946 9021032092 4184135451 8141672262 8709640996 4942627194 4268010713 3565582056 3990227202 9629672479 9562225494 8063719941 7387743604 8626332582 9724795827 0491948007 2599977268 2446838479 6736760420 3210863789 6214713648 1679235386 7049861926 0641600752 4148154184 9842104369 0236971515 6437462044 5098570732 4290026118 3776033326 6577688567 2890587901 5343034910 8546296909 9646112408 0650855159 8182627604 7619566832 5304862135 1957275870 8933037172 5399637649 2619047076 2439871243 5168003647 5432795379 0176024596 3313312976 8491084816 4714243193 4112952159 1987277031 5777157310 9833427285 6438192990 4106836334 7860288843 7189809304 1212251562 2335367433 0327709116 3402529217 9301463150 0038197182 0736782827 4941184524 9929649083 3391300007 6514797767 9690908901 9198421396 6682735099 4912499582 4558598054 4452649848 5972134333 8237132312 3739878849 3508070650 6530541988 4858621176 1910574422 4198475025 1688541616 0582319986 9306486286 4677663519 0526480361 2590840522 0801828002 4371381567 7163679890 4851740428 6476584576 0947544200 9721607027 3606745296 9113172885 6779468266 1598219078 4028271272 8421960179 2437484966 8250156507 1140459579 8018484760 0635766742 9599040433 4297991792 0992497104

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 ≡ 50 (mod 63)
• Q(1) is not a perfect square: it is ≡ 53 (mod 64)
• Q(2) is not a perfect square: it is ≡ 46 (mod 65)
• Q(3) is not a perfect square: it is ≡ 21 (mod 64)
• Q(4) is not a perfect square: it is ≡ 24 (mod 63)
• Q(5) is not a perfect square: it is ≡ 21 (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 = 561001778 1855811376 4034652723 3328104879 8534879542 4490091928 9555132921 8761697363 5381478279 7349709138 9729914909 4454036559 6154368101 5048203780 6174998372 2622245959 7174298638 0855250533 2414308428 2209940411 0786562873 0288227377 2723169192 9397583591 7595770549 6166028769 4187320898 2308690629 8602446450 9471521798 6190526893 1787851358 4759267888 1000425236 9633710161 6418872036 5494440289 3231964217 2375307700 2195859601 7544340411 4081242240 9287821439 8180300502 6373240262 8441392264 1766334152 6262902817 2569315308 0595349879 0540630075 4452260308 1930824809 3681207813 5587980583 0755732420 6018771138 5183204386 5363420648 7039823500 9961393299 2633389096 5004608012 7780439662 0142926535 9602107394 2676983579 7001085523 3638864578 7707699363 3919910432 8269116400 6315692006 7135627659 3268340520 2504313903 3015497060 7037856236 6851605454 7622373127 9010765127 5968228590 4290595493 1029995424 6186566587 7969216299 3173968863 0014294045 7553403156 0341486310 2544880416 1972533873 5406196932 4620155388 2068738768.

With those constraints, the unique continued fraction is: {0, 17, 2, 14, 1, 2, 1, 1, 2, 14, 2, 1, 1, 1, 2, 2, 2, 1, 4, 5, 3, 3, 3, 2, 3, 35, 1, 3, 1, 2, 1, 1, 1, 1, 1, 14, 1, 56, 3, 2, 1, 17, 9, 6, 7, 2, 176, 1, 22, 68, 4, 1, 4, 2, 1, 9, 5, 26, 1, 1, 2, 1, 2, 2, 1, 3, 1, 1, 2, 1, 6, 1, 1, 34, 1, 2, 3, 8, 2, 2, 1, 34, 1, 4, 5, 2, 2, 1, 16, 1, 2, 2, 15, 1, 2, 18, 3, 7, 2, 1, 1, 1, 4, 1, 6, 1, 11, 2, 1, 1, 7, 1, 1, 2, 14, 2, 2, 11, 2, 7, 1, 1, 97, 2, 1, 7, 4, 2, 4, 1, 1, 1, 1, 2, 8, 1, 2, 1, 4, 3, 1, 12, 1, 1, 1, 27, 4, 8, 1, 1, 3, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 1, 6, 2, 1, 2, 1, 1, 105, 2, 10, 2, 1, 1, 6, 8, 29, 2, 7, 9, 1, 1, 1, 4, 1, 3, 1, 1, 1, 3, 1, 8, 3, 56, 1, 6, 1, 1, 1, 1, 59, 3, 5, 3, 2, 2, 1, 1, 5, 1, 3, 1, 1, 3, 1, 1, 1, 2, 12, 5, 1, 3, 1, 13, 1, 4, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 106, 1, 1, 1, 1, 494, 1, 1, 11, 89, 1, 1, 1, 16, 4, 1, 66, 2, 1, 2, 4, 1, 11, 1, 17, 2, 2, 3, 1, 1, 2, 18, 13, 3, 1, 2, 8, 3, 9, 1, 3, 7, 6, 1, 1, 25, 2, 14, 2, 5, 4, 5, 1, 14, 1, 10, 2, 18, 2, 10, 8, 3, 1, 2, 4, 1, 8, 1, 7, 1, 12, 1, 8, 16, 1, 3, 3, 15, 1, 4, 1, 2, 1, 18, 2, 6, 2, 2, 1, 1, 4, 1, 1, 1, 1, 16, 3, 5, 1, 6, 2, 3, 6, 1, 1, 3, 2, 1, 1, 1, 8, 1, 8, 127, 10, 1, 85, 4, 2, 1, 1, 7, 1, 4, 15, 1, 1, 18, 3, 6, 1, 1, 2, 2, 1, 12, 1, 2, 9, 5, 5, 1, 8, 1, 9, 1, 2, 4, 1, 2, 11, 1, 1, 42, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 27, 1, 1, 1, 2, 1, 7, 1, 4, 1, 10, 1, 4, 1, 7, 1, 3, 7, 13, 14, 3, 1, 2, 1, 1, 1, 2, 251, 1, 1, 2, 10, 1, 4, 1, 1, 11, 1, 24, 1, 1, 1, 1, 1, 2, 5, 2, 2, 1, 1191, 1, 31, 1, 10, 2, 1, 3, 17, 1, 1, 2, 2, 1, 1, 5, 2, 8, 3, 2, 5, 1, 3, 113, 1, 2, 3, 1, 1, 14, 2, 10, 8, 2, 8, 2, 1, 1, 1, 4, 1, 1, 5, 20, 3, 3, 2, 3, 5, 12, 1, 11, 1, 8, 4, 27, 1, 1, 1, 3, 1, 1, 12, 10, 1, 6, 4, 2, 54, 1, 4, 2, 1, 1, 3, 1, 2, 15, 8, 1, 10, 2, 28, 6, 3, 2, 1, 2, 34, 1, 1, 12, 1, 5, 1, 1, 1, 5, 2, 4, 92, 3, 4, 1, 1, 1, 1, 2, 29, 2, 10, 1, 1, 1, 1, 3, 1, 5, 1, 13, 7, 65, 15, 1, 1, 10, 10, 2, 1, 4, 1, 1, 1, 1, 7, 4, 1, 1, 4, 8, 1, 49, 1, 29, 2, 9, 7, 48, 4, 1, 1, 36, 1, 3, 8, 1, 1, 1, 2, 1, 2, 2, 2, 1, 22, 1, 1, 1, 47, 4, 2, 1, 4, 1, 96, 4, 1, 93, 1, 18, 2, 5, 20, 1, 1, 7, 3, 1, 4, 2, 1, 16, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 7, 3, 1, 6, 1, 2, 6, 10, 1, 1, 2, 2, 1, 1, 1, 1, 3, 8, 3, 3, 1, 5, 1, 2, 3, 3, 1, 1, 1, 1, 2, 9, 1, 6, 1, 5, 1404, 1, 2, 1, 1, 1, 41, 12, 1, 1, 1, 1, 16, 1, 5, 1, 143, 2, 3, 5, 5, 1, 19, 2, 8, 2, 1, 1, 1, 2, 2, 6, 1, 2, 3, 1, 3, 3, 1, 2, 1, 2, 2, 4, 1, 6, 9, 2, 1, 2, 3, 6, 1, 58, 8, 3, 1, 1, 6, 1, 1, 1, 1, 234, 3, 2, 1, 28, 1, 10, 1, 28, 1, 18, 1, 6, 1, 38, 3, 2, 7, 1, 1, 5, 1, 9, 3, 9, 1, 7, 1, 1, 76, 11, 15, 1, 1, 1, 1, 1, 4, 5, 6, 2, 1, 138, 1, 2, 1, 5, 2, 3, 1, 3, 9, 1, 1, 2, 18, 4, 14, 52, 50, 1, 2, 1, 10, 4, 2, 2, 12, 6, 4, 1, 1, 1, 1, 4, 2, 2, 2, 10, 8, 1, 16, 1, 2, 5, 1, 2, 1, 5, 2, 1, 1, 1, 16, 13, 2, 2, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 15, 1, 1, 3, 1, 5, 7, 2, 13, 19, 2, 1, 12, 1, 5, 1, 159, 1, 1, 37, 1, 13, 1, 1, 825, 1, 1, 1, 9, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 2, 21, 1, 4, 1, 27, 9, 1, 9, 2, 1, 2, 2, 6, 2, 1, 6, 6, 12, 1, 14, 1, 3, 2, 1, 2, 2, 1, 16, 1, 1, 1, 1, 1, 13, 1, 378, 1, 2, 9, 1, 2, 1, 1, 7, 31, 4, 2, 1, 8, 1, 3, 2, 1, 2, 1, 1, 11, 1, 1, 3, 1, 20, 57, 3, 46, 2, 1, 19, 3, 18, 1, 4, 7, 2, 1, 2, 1, 5, 1, 2, 3, 1, 2, 1, 1, 5, 1, 4, 1, 8, 4, 1, 1, 7, 2, 1, 169, 49, 7, 6, 22, 1, 12, 2, 2, 1, 3, 77, 1, 6, 1, 6, 7, 2, 3, 3, 1, 26, 1, 3, 5, 8, 1, 4, 1, 2, 1, 3, 1, 2, 10, 5, 1, 1, 2, 11, 1, 3, 36, 3, 6, 5, 1, 28, 4, 4, 1, 3, 1, 3, 1, 1, 1, 30, 3, 13, 1, 8, 4, 1, 2, 4, 2, 2, 1, 1, 2, 2, 17, 1, 1, 1, 5, 4, 6, 33, 2, 8, 1, 5, 2, 6, 1, 2, 1, 3, 49, 5, 3, 62, 1, 2, 1, 1, 1, 3, 23, 6, 2, 10, 1, 4, 5, 2, 2, 85, 2, 7, 2, 1, 5, 85, 1, 4, 7, 3, 1, 7, 1, 463, 1, 6, 1, 2, 23, 1, 1, 4, 1, 1, 3, 1, 1, 18, 2, 1, 1, 1, 1, 2, 1, 13, 1, 1, 3, 24, 3, 7, 64, 2, 5, 1, 7, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 40, 1, 1, 12, 1, 1, 1, 3, 6, 5, 1, 2, 3, 3, 4, 1, 3, 4, 51, 2, 26, 1, 1, 2, 6, 1, 2, 4, 6, 1, 7, 4, 7, 2, 11, 5, 2, 2, 7, 1, 2, 2, 16, 1, 4, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 81, 1, 1, 26, 28, 2, 1, 11, 1, 1, 1, 35, 2, 2, 5, 3, 3, 1, 1, 4, 3, 4, 2, 1, 2, 1, 3, 2, 4, 1, 1, 1, 13, 4, 1, 1, 1, 4, 6, 2, 1, 3, 1, 3, 2, 4, 1, 34, 1, 5, 11, 2, 1, 1, 1, 3, 27, 1, 2, 4, 1, 1, 1, 7, 1, 1, 21, 1, 6, 1, 8, 2, 29, 6, 2, 1, 21, 5, 1, 1, 2, 2, 2, 2, 1, 3, 4, 1, 1, 16, 1, 4, 2, 5, 14, 6, 3, 1, 2, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 59, 1, 1, 7, 1, 2, 1, 31, 3, 1, 1, 1, 4, 2, 2, 1, 16, 1, 13, 1, 3, 1, 1, 10, 2, 6, 1, 4, 6, 1, 1, 2, 4, 1, 1, 1, 11, 2, 1, 828, 103, 3, 1, 5, 2, 1, 3, 4, 1, 1, 9, 6, 2, 2, 2, 3, 37, 1, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 2, 1, 4, 2, 31, 1, 16, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 8, 1, 4, 1, 3, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 5, 2, 2, 8, 1, 3, 1, 5, 5, 1, 11, 2, 1, 7, 1, 2, 1, 1, 1, 12, 4, 1, 4, 28, 1, 1, 6, 13, 2, 1, 5, 4, 1, 229, 1, 1, 4, 2, 1, 8, 8, 1, 14, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 6, 1, 9, 1, 2, 5, 1, 1, 5, 2, 1, 1, 3, 5, 2, 70, 2, 1, 2, 1, 2, 7, 3, 2, 1, 19, 1, 38, 1, 1, 1, 3, 17, 1, 37, 10, 1, 4, 10, 1, 5, 2, 1, 1, 13, 15, 1, 13, 1, 1, 26, 5, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 13, 1, 1, 5, 3, 1, 1, 3, 2, 1, 186, 1, 2, 1, 1, 31, 1, 1, 4, 1, 1, 1, 3, 9, 2, 3, 6, 5, 1, 3, 8, 1, 2, 226, 10, 3, 17, 1, 8, 7, 1, 15, 8, 2, 2, 65, 1, 2, 2, 2, 4, 1, 5, 1, 9, 1, 6, 1, 1, 1, 1, 1, 1, 4, 1, 117, 1, 3, 1, 2, 3, 1, 14, 1, 17, 317, 1, 102, 9, 1, 2, 1, 3, 3, 2, 1, 1, 1, 5, 2, 7, 2, 2, 2, 19, 1, 2, 1, 6, 12, 1, 1, 4, 4, 5, 13, 4, 4, 1, 4, 2, 5, 1, 1, 2, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 32, 5, 1, 3, 2, 1, 1, 5, 16, 14, 3, 1, 3, 11, 3, 1, 1, 3, 2, 4, 71, 2, 1, 4, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 7, 5, 1, 8, 1, 14, 1, 1, 2, 1, 1, 5, 9, 2, 7, 1, 9, 8, 5, 1, 1, 1, 9, 1, 23, 1, 6, 1, 6, 8, 3, 1, 15, 3, 3, 1, 1, 4, 6, 2, 1, 4, 2, 1, 2, 4, 4, 10, 1, 5, 9, 2, 2, 3, 1, 2, 8, 1, 4, 1, 9, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 5, 28, 3, 1, 1, 12, 1, 2, 2, 5, 1, 7, 21, 6, 1, 21, 1, 1, 1, 21, 1, 2, 2, 1, 1, 1, 1, 4, 1, 2, 1, 4, 2, 13, 1, 1, 2, 1, 5, 1, 2, 1, 12, 1, 4, 1, 22, 1, 1, 1, 2, 1, 1, 3, 1, 1}, giving these values for u and v:

• u= 29364534 2412807474 0503572821 5080778824 0555211578 5853876536 2458680187 5072383170 0845414268 8183309098 6007529507 4665318182 3986152879 6594870133 3092877991 5521658642 2388697771 0852722123 3229281518 6338403430 3564383422 6982821715 1645902496 1091413244 3581606110 1146505982 2159390534 2857259595 7016950990 7311872016 8936683662 8629356211 5204159754 9060309704 9011949459 9096688285 2070783494 5823176212 8623634672 9522110748 9870261534 3793178738 1739540555 7711556523 9046362987 6993421352 3937685703 3644365026 6783074598 8781007456 7721859915 3850832133 5367456654 6701393888 8840987236 9191554453 1999098515 4953595443 4867990571 0332748457 9558851375 7430961598 1545027524 5199712753 7381202728 2832451819 6224901794 1933099841 5389063732 1056929206 4526852655 4414919154 5174582278 3195798809 8152825987 3612997254 7944969788 6558672227 6535347361 0279441807 1324205166 9836395582 8660933645 9786363471 9768091533 5346176550 2331097825 9826931420 3966071599 8917201189 1694513482 9853316084 6116356855 8330443704 2436189575
• v= 513397078 3530330367 3554177894 5822898527 1663040792 4596603099 3977021983 5734796754 7689644672 0683688514 2807159599 7843253331 9131969656 7443312466 7795233120 2168498964 9746684371 1869462284 9355436905 0476357267 3288782186 1271784943 1722142375 9532295158 7407054082 0037816109 6861421480 2603349410 0738767766 3689573311 3261109482 3807153444 8927151693 5298129174 9476350021 3006040540 5918965564 9373380093 4999993620 5565427558 8771830754 2967757233 7493451695 9967713685 0693513942 8956144930 3248463543 8908957974 9718945119 8106793698 5246939037 2056299123 5034782906 1683655359 5273032325 8877694385 0740351449 9776150776 5828925429 2163284760 5050635668 1864192016 6449919154 3005626000 2224163342 7703598098 8667385808 3501160762 6018891108 3751380511 9291579761 5840644531 6009858326 9793276761 1540261339 4261786697 2929338738 9172101523 1321372533 0938496183 7793029814 7244448978 1833425781 3392976492 8585124641 6435028954 8461880228 7852754161 4327488041 3341023303 9794375122 2043705421 7707818415 1363303738 8553379162

… as taking one more term in the continued fraction would give a value of v of 801560979 3296621968 9692605115 8943214827 1207903669 7664957421 0954203063 7209349594 4461311437 2045488504 3053234888 4134843483 5852646948 3177002904 2019415683 6672346943 4478240470 9829726144 2475296047 5761181347 0244858853 1657320280 8588995635 7329770301 9665844343 0021056818 2879790475 3505835284 4207124134 8524790665 2874135063 2953816349 8581507243 9098201946 5523343967 0303768465 4789960934 7324639582 1678154042 5025318377 3825862065 1074778209 2698730195 7301730505 7147405585 2117671918 0699624827 7426636411 3679779827 3388284283 3898747281 4737154799 5381580915 8446938466 6788701934 6178126620 6425978210 5929193928 2632671898 5780697561 6201776761 8364758488 5793004657 2558483025 0831792830 7153831393 3260873437 3333839278 0057753186 1032469714 0179631297 3021715922 6691120402 6546579232 6678112373 3463684645 4108158152 3665250055 8342243974 2438510118 7770517464 7293480012 2680084823 5152149729 0666788130 6190483455 7028866863 7794657250 9870610872 0375930137 0266955121 7141772630 5164006140 8819066628 0931444719, which is too large.

We also need to calculate d = floor(c₄·v/F + 0.5) = 1571 6000045562 2892464788 1185757740 0951840221 5991629828 4107513926 4970921922 1602622896 8805496729 8115166812 5654429331 8013426644 1932750459 9436505446 7471464875 1043882588 0909239097 9428116721 2203966843 2707336674 4685182228 3892363118 5682227222 7138461890 3296525658 2359323026 4616003876 2919911410 0711271313 5842737850 4201258031 1609103582 4698730372 0968833539 3988085953 2385048777 9513636183 6363261791 4969113695 6186408558 3709397204 5432982981 6002123750 5784950426 8280042574 3884359012 4303287391 1625698073 4632677333 5409555777 6295075352 3147023089 4233294004 6785413446 3869741501 1058917751 6492784372 3357766194 4976405807 7880709163 6904164776 0675180990 4151413281 9707352133 6763268786 7135621615 8296878410 3112049211 9379125861 2770079597 1062630203 0030909127 4191807207 5767413249 1106266196 3364384660 0504902750 2819307311 5644923571 5067981913 9865173339 9098707773 1218209863 9914321391 0405784124 2265966616 1469718367 7826971869 4394278704 9237464882 9393135238 6041617006 5449733483 4344616764 9473877315 4235785329 8654618041 8398214801 3731929843 8018615091 6827272905 8933434556 4386664246 4705021137 6430344894 7234414359 6867974161 2091722276 8871930467 4516535834 6690548112 1982810810 3606101421 6973818280 6643217458 2030322860 5340297287 2884406676 1174624221 8751282940 7005043774 7921425670 0326432209 4076353959 7976545436 5033582664 5600238591 5879218956 5961242817 3123203841 4261562416 9748795565 3791167849 3011894851 9097481750 6870276189 2456384579 5095294808 1446836786 4856146350 3224258379 3839381438 0063252501 9673954031 8104796099 6546491683 1106824105 6946559674 2784599379 7148618256 5831364099 0502207702 3624266338 5949214929 4419274461 3801782751 0834050067 3000996174 5748632880 0340751925 8371060727 0774482066 9360954180 0166118008 8660894467 2434658063 1657982266 2057228932 2260402413 5183443360 4284007020 6747388967 1328505322 1203230126 7056011951 0939832646 0552334292 1433156457 3893528331 6265572130 4279022289

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= +513397078 3530330367 3554177894 5822898527 1663040792 4596603099 3977021983 5734796754 7689644672 0683688514 2807159599 7843253331 9131969656 7443312466 7795233120 2168498964 9746684371 1869462284 9355436905 0476357267 3288782186 1271784943 1722142375 9532295158 7407054082 0037816109 6861421480 2603349410 0738767766 3689573311 3261109482 3807153444 8927151693 5298129174 9476350021 3006040540 5918965564 9373380093 4999993620 5565427558 8771830754 2967757233 7493451695 9967713685 0693513942 8956144930 3248463543 8908957974 9718945119 8106793698 5246939037 2056299123 5034782906 1683655359 5273032325 8877694385 0740351449 9776150776 5828925429 2163284760 5050635668 1864192016 6449919154 3005626000 2224163342 7703598098 8667385808 3501160762 6018891108 3751380511 9291579761 5840644531 6009858326 9793276761 1540261339 4261786697 2929338738 9172101523 1321372533 0938496183 7793029814 7244448978 1833425781 3392976492 8585124641 6435028954 8461880228 7852754161 4327488041 3341023303 9794375122 2043705421 7707818415 1363303738 8553379162
• z₂= -971 0294472062 2748525673 4948189266 1610502192 9668497048 7967785830 3690164345 8043204157 4444267326 5252641481 1064190808 9313465456 7719655551 9771561493 4957462273 9739944485 1807293379 5700272458 5905280604 2442139889 9790068863 5507721111 5574464822 1423575259 9730299169 9546339903 3867828968 2673819051 8182709985 2893461524 7996244084 9556671236 2900926386 5123209208 2021822869 4036516243 1978323096 5401337514 0439122592 0664195833 6778131499 8969729203 6483021171 1978099094 5895553166 6625357799 3637537953 0475426569 4807282842 4713867444 7703398761 8069174183 6499611131 6638859871 4908415686 2096867090 7824315331 4710844880 9341886984 7421004723 6643619178 8477661679 2789684664 3770089187 5977693386 5702581851 4118840564 7626061284 5741047507 4458375749 0998444968 3920747879 4321071774 6273240773 7304852079 6866922090 2773754974 4285733729 1951663846 7466921198 4379588472 8759987350 7885205990 7344290392 0185343706 8483809113 5659139816 1001347470 1381020248 2248222967 1733163953 2477939740 0932588148 1630018505 0586100461 8401460796 0829000575 9698375706 6658033207 9503028932 0614593406 5396812845 0215270659 8141704827 5620488103 2781480074 8811614827 7198356103 9500798371 4556229546 7112920467 4354498319 4830022421 8632717420 9310709899 0639983667 2260741264 0143518252 0839239538 8013258114 7460951200 8785515468 1604509121 3222472728 0701320364 6211817866 0030085707 0991540530 3260080759 1200730815 5138624068 1845307408 0919437908 9752366334 1048693877 7037910407 1009250286 7059252820 5862829484 9353716960 9663570627 5862402930 9374266314 1012514754 0348342482 2662016331 3219738171 5927999136 5805595652 0303353554 1934697146 9505714718 1909211423 5592892422 3048572987 0815083887 2864424966 9080262101 5818597250 9863211924 5880042599 1536930164 8513755206 5445245381 9066765103 5709264283 3914328517 8096220915 4260991176 4775144985 0787069944 0034751726 7170024028 4565040969 9541303098 6290632796 6561443411 5030546137 0493465002 6068541435 1354645464
• z₃= +10 9897947562 3667594692 3224513072 1680808731 6405920102 8666746656 6242347735 7945560807 2005376767 6534898440 1126797505 2435859692 5359364749 9739808105 8250385505 2083556875 9622136686 6081236495 6563693099 2638201550 5014094675 4162830531 1229649218 9065738845 3376189440 0303853264 7478489669 0080379089 4453593487 1765369400 6589354024 8260640932 2738683095 1956311934 5330751019 3575197224 4774185944 0586360156 6228557540 3581676217 4723915598 8751211440 6849859865 8195102665 7773084095 3313540911 4265862011 7681880585 0749336848 4204796669 6632657329 5297645872 4032289631 6973184384 2291271948 9803486621 0444458068 2184494486 2009828490 9810051928 3236857960 2388778272 5217936134 3783403394 8629795302 9030664585 7310195661 8716231727 9875860976 7362139326 0054774784 3765519732 6511025858 5837408556 1573249872 8649365363 6327706213 3717401301 8637841878 4581236633 6647566387 8413481965 3420573141 0699245312 5020063296 6016794221 0725047042 9476868242 5122041496 1329619477 2748222391 3721483336 6090948913 1772908325 9385146094 9359066008 4424160855 6343941691 7867888619 4628182448 6925416568 7966701715 1090795090 2942625044 8443107083 5660581075 1157292149 3114132140 6713509565 9472707512 1071976263 7462774169 9969771923 1320941385 9302638155 8110054276 1669286049 9328872971 7459054293 3754828055 5982268715 5061712199 0997690045 4370979641 5320809137 2733170017 2724632367 2268635480 5692252591 7869010769 6191194342 7635268528 6054258098 4367408628 2416123343 8802447305 7075100073 4027499968 2692527986 4897619114 9407265606 5546121724 2578604169 6668787847 8908438777 7795221071 5447986372 7320611510 7053021943 5603432675 2949729938 8202022523 0055416831 6743595745 6368000784 5225921719 6892455958 5234145326 6627137940 0362329471 7625816888 6284279331 0505697618 9214203029 4303160559 6435399345 3435466269 3338231794 8517988644 5162440333 3877132654 6707062394 9339204350 9622571720 7086584989 0946279956 9681261007 2456406085 2995496328 9360457828 3620053133 0637151824 2829756196 0614761464 3337722075 4192334519 6586708052 1534425112 4140949853 4858174816 1569549820 7824102778 2384065235 7746962568 9859034740 7079668576 0626123386 0253094118 0309405795 1419186392 7299427074 3713147862 2643699567 4131375591 5254047118 7642177163 9164571344 5226951509 8489191671 9655072589 6101025642 9273153278 7857723816 4473763583 0482207774 7443740440 4045934095 4888641969 6677263556 5719087090 6527624643 1984301398 1121970848 2068420836 1070909725 9988242983 5871807394 0075328374 0098132578 2255034354 7416715624 6840766993 8496510047 2792635047 3112666660 2287509069 8035080520 6938958431 2127734710 0669031538 4878961400 0849651684 8272929917 7697540929 5392362449 9778925500 1713472017 2948089938 5762430473 1861089282 2834490173 5290475969 9580661973 8762674042 7010948737 4205102603 7258895214 5112756785 6103061116 9540846907 8135927328 7121607018 6587013428 3567384691 8613927202 1188789790 9246394252 1275450742 3544266791 1783396555 9480732856 6328234011 9295381958 6715283563 3220336831 3161804073
• z₄= -1571 6000045562 2892464788 1185757740 0951840221 5991629828 4107513926 4970921922 1602622896 8805496729 8115166812 5654429331 8013426644 1932750459 9436505446 7471464875 1043882588 0909239097 9428116721 2203966843 2707336674 4685182228 3892363118 5682227222 7138461890 3296525658 2359323026 4616003876 2919911410 0711271313 5842737850 4201258031 1609103582 4698730372 0968833539 3988085953 2385048777 9513636183 6363261791 4969113695 6186408558 3709397204 5432982981 6002123750 5784950426 8280042574 3884359012 4303287391 1625698073 4632677333 5409555777 6295075352 3147023089 4233294004 6785413446 3869741501 1058917751 6492784372 3357766194 4976405807 7880709163 6904164776 0675180990 4151413281 9707352133 6763268786 7135621615 8296878410 3112049211 9379125861 2770079597 1062630203 0030909127 4191807207 5767413249 1106266196 3364384660 0504902750 2819307311 5644923571 5067981913 9865173339 9098707773 1218209863 9914321391 0405784124 2265966616 1469718367 7826971869 4394278704 9237464882 9393135238 6041617006 5449733483 4344616764 9473877315 4235785329 8654618041 8398214801 3731929843 8018615091 6827272905 8933434556 4386664246 4705021137 6430344894 7234414359 6867974161 2091722276 8871930467 4516535834 6690548112 1982810810 3606101421 6973818280 6643217458 2030322860 5340297287 2884406676 1174624221 8751282940 7005043774 7921425670 0326432209 4076353959 7976545436 5033582664 5600238591 5879218956 5961242817 3123203841 4261562416 9748795565 3791167849 3011894851 9097481750 6870276189 2456384579 5095294808 1446836786 4856146350 3224258379 3839381438 0063252501 9673954031 8104796099 6546491683 1106824105 6946559674 2784599379 7148618256 5831364099 0502207702 3624266338 5949214929 4419274461 3801782751 0834050067 3000996174 5748632880 0340751925 8371060727 0774482066 9360954180 0166118008 8660894467 2434658063 1657982266 2057228932 2260402413 5183443360 4284007020 6747388967 1328505322 1203230126 7056011951 0939832646 0552334292 1433156457 3893528331 6265572130 4279022289

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= + 1540191235 0590991102 0662533683 7468695581 4989122377 3789809298 1931065950 7204390264 3068934016 2051065542 8421478799 3529759995 7395908970 2329937400 3385699360 6505496894 9240053113 5608386854 8066310715 1429071801 9866346558 3815354829 5166427127 8596885476 2221162246 0113448329 0584264440 7810048230 2216303299 1068719933 9783328447 1421460334 6781455080 5894387524 8429050063 9018121621 7756896694 8120140280 4999980861 6696282676 6315492262 8903271701 2480355087 9903141055 2080541828 6868434790 9745390631 6726873924 9156835359 4320381095 5740817111 6168897370 5104348718 5050966078 5819096977 6633083155 2221054349 9328452329 7486776287 6489854281 5151907004 5592576049 9349757462 9016878000 6672490028 3110794296 6002157425 0503482287 8056673325 1254141535 7874739284 7521933594 8029574980 9379830283 4620784018 2785360091 8788016216 7516304569 3964117599 2815488551 3379089444 1733346934 5500277344 0178929478 5755373924 9305086864 5385640686 3558262484 2982464124 0023069911 9383125366 6131116265 3123455245 4089911216 5660137486
• b= -1942 0588944124 5497051346 9896378532 3221004385 9336994097 5935571660 7380328691 6086408314 8888534653 0505282962 2128381617 8626930913 5439311103 9543122986 9914924547 9479888970 3614586759 1400544917 1810561208 4884279779 9580137727 1015442223 1148929644 2847150519 9460598339 9092679806 7735657936 5347638103 6365419970 5786923049 5992488169 9113342472 5801852773 0246418416 4043645738 8073032486 3956646193 0802675028 0878245184 1328391667 3556262999 7939458407 2966042342 3956198189 1791106333 3250715598 7275075906 0950853138 9614565684 9427734889 5406797523 6138348367 2999222263 3277719742 9816831372 4193734181 5648630662 9421689761 8683773969 4842009447 3287238357 6955323358 5579369328 7540178375 1955386773 1405163702 8237681129 5252122569 1482095014 8916751498 1996889936 7841495758 8642143549 2546481547 4609704159 3733844180 5547509948 8571467458 3903327693 4933842396 8759176945 7519974701 5770411981 4688580784 0370687413 6967618227 1318279632 2002694940 2762040496 4496445934 3466327906 4955879480 1865176296 3260037010 1172200923 6802921592 1658001151 9396751413 3316066415 9006057864 1229186813 0793625690 0430541319 6283409655 1240976206 5562960149 7623229655 4396712207 9001596742 9112459093 4225840934 8708996638 9660044843 7265434841 8621419798 1279967334 4521482528 0287036504 1678479077 6026516229 4921902401 7571030936 3209018242 6444945456 1402640729 2423635732 0060171414 1983081060 6520161518 2401461631 0277248136 3690614816 1838875817 9504732668 2097387755 4075820814 2018500573 4118505641 1725658969 8707433921 9327141255 1724805861 8748532628 2025029508 0696684964 5324032662 6439476343 1855998273 1611191304 0606707108 3869394293 9011429436 3818422847 1185784844 6097145974 1630167774 5728849933 8160524203 1637194501 9726423849 1760085198 3073860329 7027510413 0890490763 8133530207 1418528566 7828657035 6192441830 8521982352 9550289970 1574139888 0069503453 4340048056 9130081939 9082606197 2581265593 3122886823 0061092274 0986930005 2137082870 2709290928
• c= +10 9897947562 3667594692 3224513072 1680808731 6405920102 8666746656 6242347735 7945560807 2005376767 6534898440 1126797505 2435859692 5359364749 9739808105 8250385505 2083556875 9622136686 6081236495 6563693099 2638201550 5014094675 4162830531 1229649218 9065738845 3376189440 0303853264 7478489669 0080379089 4453593487 1765369400 6589354024 8260640932 2738683095 1956311934 5330751019 3575197224 4774185944 0586360156 6228557540 3581676217 4723915598 8751211440 6849859865 8195102665 7773084095 3313540911 4265862011 7681880585 0749336848 4204796669 6632657329 5297645872 4032289631 6973184384 2291271948 9803486621 0444458068 2184494486 2009828490 9810051928 3236857960 2388778272 5217936134 3783403394 8629795302 9030664585 7310195661 8716231727 9875860976 7362139326 0054774784 3765519732 6511025858 5837408556 1573249872 8649365363 6327706213 3717401301 8637841878 4581236633 6647566387 8413481965 3420573141 0699245312 5020063296 6016794221 0725047042 9476868242 5122041496 1329619477 2748222391 3721483336 6090948913 1772908325 9385146094 9359066008 4424160855 6343941691 7867888619 4628182448 6925416568 7966701715 1090795090 2942625044 8443107083 5660581075 1157292149 3114132140 6713509565 9472707512 1071976263 7462774169 9969771923 1320941385 9302638155 8110054276 1669286049 9328872971 7459054293 3754828055 5982268715 5061712199 0997690045 4370979641 5320809137 2733170017 2724632367 2268635480 5692252591 7869010769 6191194342 7635268528 6054258098 4367408628 2416123343 8802447305 7075100073 4027499968 2692527986 4897619114 9407265606 5546121724 2578604169 6668787847 8908438777 7795221071 5447986372 7320611510 7053021943 5603432675 2949729938 8202022523 0055416831 6743595745 6368000784 5225921719 6892455958 5234145326 6627137940 0362329471 7625816888 6284279331 0505697618 9214203029 4303160559 6435399345 3435466269 3338231794 8517988644 5162440333 3877132654 6707062394 9339204350 9622571720 7086584989 0946279956 9681261007 2456406085 2995496328 9360457828 3620053133 0637151824 2829756196 0614761464 3337722075 4192334519 6586708052 1534425112 4140949853 4858174816 1569549820 7824102778 2384065235 7746962568 9859034740 7079668576 0626123386 0253094118 0309405795 1419186392 7299427074 3713147862 2643699567 4131375591 5254047118 7642177163 9164571344 5226951509 8489191671 9655072589 6101025642 9273153278 7857723816 4473763583 0482207774 7443740440 4045934095 4888641969 6677263556 5719087090 6527624643 1984301398 1121970848 2068420836 1070909725 9988242983 5871807394 0075328374 0098132578 2255034354 7416715624 6840766993 8496510047 2792635047 3112666660 2287509069 8035080520 6938958431 2127734710 0669031538 4878961400 0849651684 8272929917 7697540929 5392362449 9778925500 1713472017 2948089938 5762430473 1861089282 2834490173 5290475969 9580661973 8762674042 7010948737 4205102603 7258895214 5112756785 6103061116 9540846907 8135927328 7121607018 6587013428 3567384691 8613927202 1188789790 9246394252 1275450742 3544266791 1783396555 9480732856 6328234011 9295381958 6715283563 3220336831 3161804073

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.