Primality Certificate for (18067^4201-1)/18066

This certificate uses a theorem of Coppersmith and Howgrave-Graham 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 28.267544% 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) = 28.267544%

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 = 1081 suffices.

Given such a witness, Pocklington's Theorem shows that every prime factor of N ≡ 1 (mod F). As F⁴>N, N can have no more than three prime factors.

Express N in base F

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

• c₁= 2995 4817311629 7199481064 9839054022 7132558562 8481949168 4522540256 2914190296 8725695908 1836722301 1985128482 2959409417 1221410946 2071497712 0620421171 4827720029 4459741348 9215925388 4366337786 7963864790 2402470906 7038736600 4537201432 7946684230 4628079245 5379016084 5328151514 3221050119 3171038373 2013597201 1798250222 8351017552 1530212849 8251468336 2204114885 9595884158 4028758641 9497135328 8125058321 9404414384 7594012474 2506377751 9078590233 2709588906 6540065503 9402210263 4868270646 0131534313 2325554815 3422842719 3652908375 2257716693 2877485766 2616895192 7072052976 9929619031 4480809026 2131631234 1862058308 1972556451 4183961742 4346823747 3678436942 0606731394 8692777802 6658551876 4350834506 0794427395 0817026563 2720925411 9069564318 3519651477 8755510674 8718098925 3714936657 6791438795 9171459869 5159150650 0641020369 8506775424 6955831185 6206875096 7143940122 5908438617 2068435638 4856643958 2669298324 5005431392 9241668967 0258813677 7767191813 2040711148 9479394780 0082422687 2395978861 6603833845 1872562858 7121865718 3975393543 0526171669 0016301445 0324371669 8392508900 7099127910 7986437047 2152114731 2992265674 9672119548 7935978470 2656791825 2893416153 2985924394 6157267712 5124718832 5149529688 9079620722 1924684133 0714721042 0183199931 9026916789 1466773945 2164840992 4152901113 6328110949 4678207678 0604552848 7789500910 7530937350 9827973537 1973319405 0096191424 2246411290 8900984318 0550363323 7611115085 4100375799 5034542571 0735147905 2264942345 1616542404 1848276162 6484081609 7575602565 2808620350 6623996006 7270369888 0582825724 3106673219 7139166139 0731543738 4648875950 9549132018 1262704558 3190673288 6195592456 1246617650 2422833394 6337191546 7713337363 0941605529 4877086790 5248838563 6664759246 7301184057 8802283427 8504749110 8871440427 4184470299 4296541360 1478498271 6036271177 9698706857 9735259789 9330811515 1159143373 9261521043 7101097542 1127270955 7851009959 1503347430 7136082988 1180636012 8044518175 0991353586 3758230066 8480591406 8616645655 5458152703 4952081711 3995102394 2238686584 9382979420 3550173937 8598660484 7484205754 5762777539 3919982681 0155882473 4081517427 1128769269 5673735937 3043583800 5866497864 0019780992 2114004943 9239680984 4588774136 7040316709 8576814761 8279706723 4524305168 2405738851 6286811027 9739571698 5377227036 3755680129 9343896638 5993285053 9790020112 7812211985 3857423685 4731130641 9908915859 1361097115 2959287908 2422861093 3099262297 3046412413 4639692782 9243630217 7228832938 5539601527 7263412814 6471853281 5652799137 2814200790 6250675427 4092445588 1571697271 3001880710 4824861494 2622376516 3372818133 1235894909 5081048199 0675903593 1233014194 5719750249 9460212967 9966798288 5380746340 9318716414 6960006701 2970518015 6975684992 5462802883 8536713344 0778759313 2618818552 1799162631 0590165860 3326513293 4060967023 9195299540 2837833241 8550805639 1587278698 9583435223 4089093844 2447756076 1156153113 5223859380 2275050901 0767477875 7484060124 2496266942 0718416249 7758872600 4545455962 4654383353 0194036884 8716225127 3292869535 8018807584 1956237631 7940106456 5066508972 0205352025 2733428691 9784166325 1033877258 4685912503 9438561561 3186390704 7178268680 6177001540 3837503742 7107482983 2585338344 4401211205 7739924885 6421311429 8518534427 6654077471 3322543436 1408091364 1490273010 4893914025 8071890082 8104675815 0590999172 7775117707 2494900612 1302613994 5716984710 1207528757 8881742179 0597832737 1265516152 6975435063 0940767827 9653141279 5010965109 5534610515 7057979147 4272188796 2988723362 3367576453 2856199372 8230377344 1558174475 8526301780 7707140924 5561742785 4100054273 6895839647 4285379504 0142155691 2857110439 6956688430 1657302484 2383490878 5610390713 7618846031 0681282605 8987731650 0755440166 8865002408 6142665096 5587906497 9314227304 1761450757 3369993234 2582773940 8185699185 4778134938 0440655270 5682971187 2459705859 2343389653 2515639137 6140799644 0859400622 4022979266 0924514090 4274910237 6706815608 2186171109 7305494777 7112384321 4518596730 1471527974 4211996156 3045673724 4971340582 2530119515 2058535873 5350441640 0802939771 8024510573 8736386996 5560520213 4455647075 8421586019 5593586036 2019682187 7959166278 6067504028 9445001258 5364061917 1995766962 8134816607 0332738105 5408292230 2577243977 6631600815 1084287589 4401243312 4681530942 9117205038 2261648672 0290369913 7729117779 0422047355 7006196743 6875489351 6012885233 4450007879 4902278690 8261638063 2847568512 2078052417 8803348554 8372701493 6637917289 8729824198 1607662255 0868028786 7873208074 4061462225 3929478886 9290090541 8884511932 2736982912 9153867343 4938422795 0323441368 6704117969 8707152959 9149988364 4000054410 5269245675 2513629511 7885898050 5093127059 0553114169 5910316260 4758871359 7529745597 0094988806 6080080792 9625650181 2389569945 0089546177 1951587224 6685966961 8440015325 2713409585 7421119586 8180515607 1122299608 7393247085 4571134338 6929040337 4748867589 3453945989 8346444880 8629502760 9881599512 7844814483 0324195285 1485781671 0044289658 3262005948 3388835508 0630219148 5027267612 1451731163 6967121226 8476298187 4474848404 9075613023 0709314150 6918450398 2698716442 6640909272 6103623889 3211997834 7587866222 2499352558 4290121277 1007650823 0442633148 8183605057 1157973857 3437213159 7030017655 2479605204 8356092477 5834066120 0379607834 2171969460 3889627253 7786677266 9654221572 1090506417 8150639807 7005689151 7826821690 6102142563 4504458737 2231436664 2972772680 0355741707 9492490994 9006961411
• c₂= 11 4113178184 8116266288 3480287339 4037636769 3176064596 5364991225 9698271378 5796906185 5395784980 6082820022 9912697104 4710577653 5645683019 6770220295 6061970347 5106619543 6838877317 9730046397 0917962409 8208531499 4431340444 8428155756 8048291433 9170676742 5088192776 9464180272 1575549476 6689046884 2752684415 6595427770 9296339279 5206058941 8201598648 2654072081 4143480847 9376196649 2628296243 9283403708 5232798436 7659827797 8271873479 4712063615 4143299275 5989864948 6301009946 2161981118 4004240045 6668520140 0108710475 8989290890 1623801740 7890595272 5562882060 8610280520 6819038164 2057154384 2856020518 8984765926 7373405274 9435160992 5694888688 8122608368 2047715539 8103057114 9583019569 3588037169 2437672451 1190142427 2082660379 2446959708 2303783640 2185852594 6933906794 7785227903 2383281458 3980110099 7339115137 0101654300 8566856219 6414269330 5402920872 4423004960 5600903677 5580583644 5411894474 1722243213 0753017715 4779647819 4896952376 6384128748 5057746755 2459285072 3414042026 4287623032 1378151977 3015515529 3414829760 5971800615 9611783681 1063191016 1418884529 8988111602 3846207745 9516771629 7331523340 8521839073 8584108405 7785947803 0771911588 9777388502 9724077268 2995143583 3653669223 7295107793 5269575062 1169252739 8806148036 2746932546 1318717258 1936189310 0793585857 9840632592 5374715212 2276441771 3269907455 9025041557 3340364103 1912682470 8536426573 3002269165 8365884487 3252238960 4494774221 8237713903 0278276370 4663122104 0811207781 2411598202 5643382234 9016334594 6215230519 3832828290 0701937259 8681299171 9165823721 4818142976 0286656572 6570554264 5405683986 5035805749 2002771440 8118824426 0882472822 8422558885 8104486377 9182446036 6315581499 7966367868 3727106489 6943625601 5805438940 6448072388 3696865057 8788643751 2405259838 2279812724 2410856138 8850127398 4585664002 6091671523 9381251412 6553880449 7929327393 1500927974 1737377455 5511772069 2300608503 7462572767 5651816489 0675044621 8877507107 6054120772 7026466611 2910454467 0634224491 4375075145 5371321239 4204275731 1779550226 5604641622 2315470953 9387469859 6471521611 8401724020 5331402780 4779608447 4114108112 1038923228 1479370799 6000461164 1775423209 5006139906 0776919782 0387398879 7534144471 6041042696 8506218080 5056199566 9207438821 5665064850 1121003217 6343327094 1465915598 1172069753 3384854249 8532282624 2936473185 1227859567 5925377785 3691873077 1984601830 0834850174 3824765189 1063060686 9508746911 5176012898 4069784498 4693516823 2312287015 8440315499 6224197767 6403352402 1973210248 5904632873 9868827380 3979697349 5188627673 9563807162 5485478827 4899964689 2091127352 8496634317 5578270654 2896977405 9547602449 2247113488 8791663188 3071131036 3180599082 0329562519 3840149716 7511721159 6295236401 2605134660 5277232596 5774162872 8961766801 2738039318 6050042855 4863091665 2844632219 9056024794 4891500693 4027529546 5798564245 2225000792 5460552452 6926603574 0704105756 9249701227 1238602396 1751558986 4512795252 9228671882 3379608390 9431843993 6336063495 5016364454 0382742647 7867274808 5301053986 6677038455 1398170396 5718680356 8332263866 9947918349 4183538490 7316633561 1976120946 8439854356 9182863067 8722548721 4278528736 7043353339 8257410710 3496917039 6295574835 1059280022 0555692837 8888512219 9388651190 6990473517 1268119937 7778182835 9138903159 2570254002 9162244961 1746079529 4057297412 6487291852 7756132477 3013309000 9120126180 7938269745 5787944237 2228081108 6039993169 4236648738 7379836873 3741813851 8670473406 4435188762 8840714902 4351271123 5740746146 5051179529 7510318631 6310850556 4461906336 4331681208 9856944173 8834839415 7221052125 7110541790 6895746031 3463748794 0559335231 8231850272 6835763718 3645458657 4436506682 9581403520 5107702922 9539714972 5006651493 8080911218 5998933314 5827408949 8135658291 9425380680 9097370854 7769350728 9835408683 6531152500 2953124021 9242063720 2942723876 4339945471 7103956148 9800244881 6967308653 0006024981 8422849428 5407721659 0159761170 0216193604 0851060434 5711560464 1494240054 8904663853 3981385398 7554148042 0361865089 1675075902 1324512686 2794890210 8364812027 7679856392 7139611605 8087937357 9148719132 2592071388 7276293475 7223435879 8736602797 1800161474 0723291797 2150008655 6429282532 5886158658 5589376067 5690986069 9199005515 5712494649 1162843548 1282477937 9752188013 1645030759 4334090238 6005493662 9078198102 0283612312 7845463949 3163193861 3809796465 8746165456 7558068120 9070007695 2999133220 5698346874 3984247598 1204092733 3513652774 8474276153 0462678704 7324182601 2722717915 8460076159 6214492623 8344077598 4858154459 0313384517 8994823670 0003436586 7233276587 7017874341 4110413115 7139970373 0926949882 7243201951 1268031620 8008569197 6599281358 1979822945 2652063287 8050856922 7664350468 0208522357 8454276775 6011300762 6453601040 2869922496 7994358367 6388755724 3723928420 8439326901 4304521525 4439770216 3671737571 6038066391 0880572151 3579306990 8433841927 3197196657 0454708238 7945301564 2745385360 8952588633 0256246460 7819058978 2249944889 5621801435 1609961762 1901568382 8907920870 6481908753 1670037978 5709501060 8911264528 3229824836 0290501407 1476909074 8240603174 9993564486 8537632531 9124263605 8128932270 0868363977 2745835382 9818410791 8694789353 2514461514 2672795096 2094922102 9493834980 1388809372 2480050813 9010807394 8484996756 4228594275 8499704463 8873728696 3367113146 1687397899 8934176477 9726738335 4848147362 6528781142 1629906647 1506843788 2061274085 9394455071 1368919010 6497094449 9755681195 2822789984 6356680012 8981276178 7664318976 4282554940 6098509469 3248171190 4996697591 1006453445 9853704935 6023886194 6751348527 1447185858 3947227624 6081044795 1938828783 0901470846 7871189262 9010541511 1757938866 5431480082 6671943600 8948818158 8820612399 5335435308 5325402752 1858566247 7646823590 3902081845 2111584372 0023362643 8771529649 6701790363 2163634862 8635429147 8060365523 4979091874 6499119366 4485865839 2514206436 5880800059 0644782335 6314053113 4892808337 3971926284 2308063025 2105337737 0152296911 8469236185 7648900896 4117288989 5113052492 7614006558 9772427970 9372109489 7878528516 7877689985 5080991032 2358964053 6084221461 1601603097 1454917788 3366379440 3624464730 4063719742 5309389074 7299904615 6829675392 4573973788 4143638593 7633402394 4951687616 5795807643 2106633782 7437624235 9316323374 0518784604 3032604037 5208701415 0430988873 3658215223 6689842275 5495909055 1886093985 3272938927 0960118626 2689251143 6073144068 2406719844 2759048763 4661379547 2745387985 3536171972 7921333248 8114304964 5048357840 5029822172 1510109371 6796731289 8325701025 4288570516 9880583622 8720892029 5605128539 8257731292 4022320457 7882628286 8694902020 4266644296 2466276298 2564358763 7748506270 3818120064 4761106729 8340165605 4934575737 6537343620 5960120257 5241208846 1751540579 3246228955 9583228608 5829205951 3107784876 7937145069 7622898984 5508442652 9975551514 7718080115 1542723305 3434707741 7528999017 9414475240 3005288440 5681271786 8793254246 5086863802 4138322119 7756415039 2411118360 5458785299 5356679335 6901653743 4736693777 4839365093 9856620647 7599567301 1515176034 3768739074 2582003200 7706525301 5642570686 6833147039 4338661618 3385857137 1397726856 7171451098 2574494263 0710943834 0103975871 9080531340 8054445716 2889285184 6744621552 2609830281 6447842196 2930879504 2912684843 0035060405 1660929530 8620733500 4252495165 5478539156 4907433176 9575834688 1384566062 8064603860 5554717260 4504758581 8903712944 0902436077 9816470590 9877082259 5828541544 9851275277 0696304589 5174817886 5104048841 7962822285 7827940135 1900142796 9839791048 5930431547 0503146931 4782784034 4195026874 3450547536 6525525470 9608424399 5549818581 5570969799 7504561604 1804909475 3384394356 8841025532 2040950984 6600035019 0497746611 1616971824 9068606752 7800972766 5682277571 2726004706 0406732306 4657356490 1975211338 6649444919 5023341251 3010547084 3748661041 9249101883 8973275919 4018841930 0091969050 1108343533 5799994575 8222966293 6455209574 0899517262 2757214623 6796069475 5504258017 5678857699 2563836710 2340436662 1851887122 7422440360 8588002708 2821076287 6745783404 5629601954 3903052314 5254123257 6506762694 8111304280 4562364644 2231043722 7086845788 8957436716 8605060840 9180193408 0957143404 1572777851 5943245140 8571566657 2413310458 8930850883 7074995134 4058400792 4793082407 6104704169 2102504931 7936722331 6691147440 4844754717 6921795992 2851245541 5898574587 5948465343 6113922280 1496082672 0529420717

Brillhart, Lehmer and Selfridge

Brillhart, Lehmer and Selfridge's Theorem shows that N has exactly two prime factors if and only if c₁²-4·c₂ is a perfect square.

Here, c₁²-4·c₂ is ≡ 21 (mod 64) and therefore cannot be a square and this stage of the proof is passed.

Here, c₁²-4·c₂ is not a square: Its root s is 2995 4817311629 7199481064 9839054022 7132558562 8481949168 4522540256 2914190296 8725695908 1836722301 1985128482 2959409417 1221410946 2071497712 0620421171 4827720029 4459741348 9215925388 4366337786 7963864790 2402470906 7038736600 4537201432 7946684230 4628079245 5379016084 5328151514 3221050119 3171038373 2013597201 1798250222 8351017552 1530212849 8251468336 2204114885 9595884158 4028758641 9497135328 8125058321 9404414384 7594012474 2506377751 9078590233 2709588906 6540065503 9402210263 4868270646 0131534313 2325554815 3422842719 3652908375 2257716693 2877485766 2616895192 7072052976 9929619031 4480809026 2131631234 1862058308 1972556451 4183961742 4346823747 3678436942 0606731394 8692777802 6658551876 4350834506 0794427395 0817026563 2720925411 9069564318 3519651477 8755510674 8718098925 3714936657 6791438795 9171459869 5159150650 0641020369 8506775424 6955831185 6206875096 7143940122 5908438617 2068435638 4856643958 2669298324 5005431392 9241668967 0258813677 7767191813 2040711148 9479394780 0082422687 2395978861 6603833845 1872562858 7121865718 3975393543 0526171669 0016301445 0324371669 8392508900 7099127910 7986437047 2152114731 2992265674 9672119548 7935978470 2656791825 2893416153 2985924394 6157267712 5124718832 5149529688 9079620722 1924684133 0714721042 0183199931 9026916789 1466773945 2164840992 4152901113 6328110949 4678207678 0604552848 7789500910 7530937350 9827973537 1973319405 0096191424 2246411290 8900984318 0550363323 7611115085 4100375799 5034542571 0735147905 2264942345 1616542404 1848276162 6484081609 7575602565 2808620350 6623996006 7270369888 0582825724 3106673219 7139166139 0731543738 4648875950 9549132018 1262704558 3190673288 6195592456 1246617650 2422833394 6337191546 7713337363 0941605529 4877086790 5248838563 6664759246 7301184057 8802283427 8504749110 8871440427 4184470299 4296541360 1478498271 6036271177 9698706857 9735259789 9330811515 1159143373 9261521043 7101097542 1127270955 7851009959 1503347430 7136082988 1180636012 8044518175 0991353586 3758230066 8480591406 8616645655 5458152703 4952081711 3995102394 2238686584 9382979420 3550173937 8598660484 7484205754 5762777539 3919982681 0155882473 4081517427 1128769269 5673735937 3043583800 5866497864 0019780992 2114004943 9239680984 4588774136 7040316709 8576814761 8279706723 4524305168 2405738851 6286811027 9739571698 5377227036 3755680129 9343896638 5993285053 9790020112 7812211985 3857423685 4731130641 9908915859 1361097115 2959287908 2422861093 3099262297 3046412413 4639692782 9243630217 7228832938 5539601527 7263412814 6471853281 5576608935 8873303936 0902212562 8901582754 0698144655 3509477806 9010100770 2502122819 9062205129 4065253066 7342834766 1105024129 2856471892 4246631571 0484981984 6563985754 4384299888 0693936338 1837735752 9237296511 7750805274 7185446156 6547038650 2020960186 9788578140 0195611829 8699308319 0766436101 8855384822 8319457158 2287560558 8415731338 1677233826 2012273082 9170796233 8749398753 8343586740 3041952343 4960471406 7286239687 2983570874 6685534314 9015363302 7004793978 3358843082 3683965799 7885325978 2770943256 4473173513 6797976867 4832401376 9003743604 3395531420 5820083860 7861351811 8690110528 1263931941 4225593625 7527472952 5440041161 0194139045 5608258372 8739258750 5971035089 7645570981 9184182722 3877759344 7664469084 5889370511 7481970933 0384959524 0276544526 2264538648 8153697573 2191105676 7998266893 2534336060 1153484192 2563916100 5227133925 2421692353 0244453250 9513697310 1820484715 9825884955 7094792521 7245474678 6948218372 3479930611 4909817997 7156272997 7106660736 9695650365 7548817104 2652642754 9886934145 0831851212 4932182142 8928076158 8844679344 3455342466 4435933411 8294192563 5636206369 1402558774 6283632597 0769060951 4840256119 5855481191 9717845586 1140224092 4663348736 1021878929 9928494261 3671008746 1115053669 9523757625 0084573152 5146594613 0298507725 1655927360 3180418125 4367834386 3217988848 8867413410 9911897599 9854789640 7481718617 5002849479 5829123805 4424080151 5653465365 2381567455 5588305086 2896524985 5717494245 2908386922 4331440673 2129925266 7204993562 7655232880 2865804505 3988413364 8832346074 5267179921 9080046780 1172790469 7888082048 5356067460 9692418681 6444016069 9643410388 0869707450 1969863947 5974343247 9670207964 6997476322 8781209965 3615811457 3662397163 7745936904 6654328244 3382453463 1194298513 1497612917 0078163878 9307476882 6964269032 0998058659 5568614334 2688492338 8266153269 6350215179 1868899054 5140164634 8242491548 4460890095 9778197761 5061456252 6042927162 4848161620 0799381582 3924568090 8426641299 6764147754 4165380835 8809005625 9722337354 1413784011 9283995333 0722547592 7511261168 3960500820 8883946325 0047501612 5905669373 4292343353 1640569837 3803626016 9358821080 8628521898 0527664917 0305376989 7396434925 6669757690 2078831773 5545439455 4926387947 8408463819 9077384865 4866455527 2426147642 3107637684 7328139174 8427347511 4863562193 1066653850 5354962476 7744550528 8941084168 5746600698 1767056692 9200518153 9895646366 8318720070 0569384444 7947889329 5576353156 1118152768 3358273143 6986392243 0182193210 9331223253 1906446800 1048962679 5638090109 0902863788 8672632134 8364883365 1900147383 9968880263 5904780951 6544102264 1181732781 0343794505 1539054881 5026774868 8885281836 7674123710 1211709229 8396308733 0483205911 2808709488 3668883322 4021381149 0639486937 6944325160 0622194254 0412946587 9576721228 3148382629 1907531276 4645385154 3731346029 5860012035 2710665678 6874960799 9556481067 1296589683 0242090270 5810885627 0798974088

It can be verified that s² < c₁²-4·c₂ < (s+1)² and therefore this stage of the proof is passed.

Coppersmith and Howgrave-Graham

We are left with two possibilities for N: either it has exactly three prime factors or it is prime. The non-existence of exactly three factors is demonstrated by the Theorem of Coppersmith and Howgrave-Graham, here performed by a Pari/GP script written by John Renze and David Broadhurst. Here is the stdout:

realprecision = 7003 significant digits (7000 digits displayed)

Welcome to the CHG primality prover!
------------------------------------

Input file is:  IO\46931069.cin
Certificate file is:  IO\46931069.chg
Found values of n, F and G.
    Number to be tested has 17879 digits.
    Modulus has 5054 digits.
Modulus is 28.26754434% of n.

NOTICE: This program assumes that n has passed
    a BLS PRP-test with n, F, and G as given.  If
    not, then any results will be invalid!

Square test passed for F >> G.  Using modified right endpoint.

Search for factors congruent to 1.
    Running CHG with h = 8, u = 3. Right endpoint has 2718 digits.
        Done!  Time elapsed:  2926219ms.
    Running CHG with h = 8, u = 3. Right endpoint has 2568 digits.
        Done!  Time elapsed:  3658453ms.
    Running CHG with h = 7, u = 2. Right endpoint has 2382 digits.
        Done!  Time elapsed:  185641ms.
    Running CHG with h = 7, u = 2. Right endpoint has 2328 digits.
        Done!  Time elapsed:  191625ms.
    Running CHG with h = 7, u = 2. Right endpoint has 2247 digits.
        Done!  Time elapsed:  179734ms.
    Running CHG with h = 7, u = 2. Right endpoint has 2126 digits.
        Done!  Time elapsed:  183813ms.
    Running CHG with h = 6, u = 2. Right endpoint has 1945 digits.
        Done!  Time elapsed:  135172ms.
    Running CHG with h = 6, u = 2. Right endpoint has 1750 digits.
        Done!  Time elapsed:  106953ms.
    Running CHG with h = 6, u = 2. Right endpoint has 1554 digits.
        Done!  Time elapsed:  88265ms.
    Running CHG with h = 5, u = 1. Right endpoint has 1359 digits.
        Done!  Time elapsed:  53422ms.
    Running CHG with h = 5, u = 1. Right endpoint has 1032 digits.
        Done!  Time elapsed:  60844ms.
    Running CHG with h = 5, u = 1. Right endpoint has 379 digits.
        Done!  Time elapsed:  124641ms.
A certificate has been saved to the file:  IO\46931069.chg

Running David Broadhurst's verifier on the saved certificate...

Testing a PRP called "IO\46931069.cin".

Pol[1, 1] with [h, u]=[5, 1] has ratio=4.882389688 E-722 at X, ratio=4.39820814 E-1100 at Y, witness=3.
Pol[2, 1] with [h, u]=[4, 1] has ratio=3.865063003 E-378 at X, ratio=1.240557869 E-653 at Y, witness=2.
Pol[3, 1] with [h, u]=[4, 1] has ratio=1.187693657 E-327 at X, ratio=2.691465580 E-327 at Y, witness=2.
Pol[4, 1] with [h, u]=[6, 2] has ratio=0.0919194979 at X, ratio=5.80826488 E-392 at Y, witness=3.
Pol[5, 1] with [h, u]=[6, 2] has ratio=0.2656821240 at X, ratio=1.246345465 E-391 at Y, witness=3.
Pol[6, 1] with [h, u]=[6, 2] has ratio=0.812812234 at X, ratio=1.010256626 E-391 at Y, witness=2.
Pol[7, 1] with [h, u]=[6, 2] has ratio=1.295427841 E-142 at X, ratio=2.090454151 E-363 at Y, witness=5.
Pol[8, 1] with [h, u]=[6, 2] has ratio=7.53270269 E-122 at X, ratio=1.069300915 E-242 at Y, witness=2.
Pol[9, 1] with [h, u]=[6, 2] has ratio=5.232373766 E-82 at X, ratio=4.83122557 E-162 at Y, witness=2.
Pol[10, 1] with [h, u]=[6, 2] has ratio=6.396694380 E-55 at X, ratio=4.513488208 E-108 at Y, witness=2.
Pol[11, 1] with [h, u]=[8, 3] has ratio=0.3057695845 at X, ratio=1.829528558 E-559 at Y, witness=3.
Pol[12, 1] with [h, u]=[8, 3] has ratio=7.20998576 E-151 at X, ratio=1.525922097 E-450 at Y, witness=7.

Validated in 19 sec.


Congratulations! n is prime!

The actual input file containing N and F and the output certificate are included in this file.