Primality Certificate for (13096^5953-1)/13095

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 28.084688% 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) = 28.084688%

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 = 5 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= 11 7842244614 0416114635 8443823584 4108267151 7634232963 5999380558 7293322119 0655425263 4120689078 8095594381 2607164298 3097709234 3258543784 4414340752 8927798681 4003899236 2458276059 1930085517 8673290797 5240095638 3983909963 2592551403 2221441624 8537530482 2520807088 6494188090 8261823294 9880833556 8746990111 5448492031 7132188802 5385825942 8271979066 8282458851 0562518267 5656940084 3363083463 8770747835 7038605156 9695778615 9153850211 1160278996 5439958329 3637672283 7794159616 3501779289 0186290266 8067885144 8739045343 6850470107 7106724485 0537822659 3572992944 6756979651 8436539616 7737109533 0243634003 3307915054 8903175275 7239025899 0494859747 1801438895 5105014618 1226380100 9763758676 7880461680 0094260774 1037072013 9435269863 6627609561 5878637793 5810435190 1602905385 2020708405 7169269670 4097262319 8069329604 6506954795 0948411141 1290524058 7002328400 5742724341 4888008456 8519657592 5355008513 1385963655 9406382186 5739266463 8925064345 7230163735 0510639541 9784347340 9630795456 7983640118 6782155692 3676302764 1967449417 4668400204 2265125468 8290747179 3522495278 8825446082 2097980524 1980293637 7438465664 0330754676 4003862592 4635923204 7552094231 1997226315 1453547716 8719715499 0443777158 3503989689 2595459597 6416043078 7006355663 1534836629 3779119651 7854403713 4460625538 8080788440 1291814676 1183466424 4964409154 5751516519 2668840004 4409353360 5236186875 5377822196 1422787131 7937271501 5448742172 2850332474 6812334903 8885775700 5662715678 2502047288 8795580527 5826568273 6861208044 2666239348 0324402189 7400457322 3925355085 8960551245 0144168263 2791928513 0698718039 2587868742 9178534624 1973005461 9904674519 7301819973 1515410386 0114797882 5246609018 1700008537 0130779212 5089799600 6464169294 6454326374 6111262002 8793093242 9470338973 2483944702 3882343186 8784477731 0908506491 7022833797 7074459641 3503186270 7343276246 2780381301 0984108244 6047418595 4707819832 0211076409 3036945241 0917970077 5750912420 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• c₂= 6 1094679983 3278233114 8144079846 8383261484 2973348745 4757181597 1479726240 4853345833 2921685324 2348040958 8472414766 7346582446 2585231034 1439220134 5830332565 1374491007 0235889481 5339788492 6723506622 2677183800 1281856175 1860717207 8451088275 1158636564 3743726943 6246513330 2038898092 4767570736 3488652772 3702964205 1639665517 9398013589 4235011977 6361566413 8924930000 4505113420 6704261168 6935699615 7555692076 2911127746 9494602751 5762727842 3347196055 1602730691 7631836908 4696061493 6601337254 7412535793 2967920912 9542840100 7671303405 8690252326 7084110276 6709458499 3792566376 9957627351 3752250829 7060253288 1486941738 0820677421 1232197932 9167757923 1163346358 1098318721 3624170882 7045163281 2323439783 3553995602 2002905006 6651679345 3790968965 4123195057 6220784312 0366457985 0309595425 0972004401 1249649521 0457859551 8637611017 7538152305 4583649427 3148900151 0808842620 9790131990 0450030588 6570615070 6092633910 2170657564 5223001720 5681109795 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9563068063 9572217236 6654344692 1064302429 9562159195 5203746041 5151959312 0087064040 1996543588 9111590168 4566179472 1087620428 0022231210 5265478357 2307741697 9784483410 7458071268 4847044278 8119620106 4906983315 1430452030 3369724326 6735276889 1347316227 6002058371 7876222740 2037750227 6088315346 3725422469 7264130457 2019071221 3700348070 4980864612 1424148039 5465518846 2636116976 4918428566 7025989444 3873098715 7370729561 3132505909 7408730041 7910612998 1949012273 4886931629 9154069526 8090823923 2466367353 0962755142 8522683788 5978764293 6413346322 9090384425 2424462961 0516420702 4946938286 8885795365 4614993699 8524882534 5462199148 7191514752 1559254736 2084218357 4645323471 1849860798 1477256113 4902876857 5840297411 0325548899 3221173512 5844500096 9551905365 2008150683 8580460030 8015618912 3900525846 1449104704 2196056531 1033453250 4080820789 4458562738 4058174447 4177813910 8048754586 0022723420 1320258494 4544182095 5489791453 4622566651 8057555932 1220374525 7682002721 3212976326 8036073435 5598416330 4690141220 2908782035 5945804856 6045346881 9506613658 9303710564 5064246744 0979766060 4485096210 0384299933 8529518540 2636580078 9328516209 2074243802 2149570450 4361527461 9258334926 3644869305 5093859515 4534823478 0389374954 1572086036 8013213680 5714318176 4244758721 2086735630 7459860705 6630805805 8897982141 7590414449 0618545592 9216580355 4893885835 7886429224 8184297083 4117751486 0934505059 3065731052 2311479376 7525052867 3897551245 8864406743 9959014314 3295558314 7013925459 6719762303 3235742473 3633104766 4291418759 6748966574 0197093437 6330424268 1507581149 5228373389 4648319486 0464609546 2818051623 7474837739 4889903881 8858084770 9763429747 0247322971 4886149197 1350225294 2941779069 6234111285 9061793460 3640622206 7692000162 3381181485 6888534190 7483753025 8085856941 1853188332 4989559848 7378522714 4380586761 9521131027 2021835490 4443134203 7243643656 2622784875 0855026221 9585039130 5819508402 3797963738 9505461001 3264323938 9724406782 6196691659 8520122822 1808038215 7791510568 2157933025 4420600438 8470915019 2450991912 1638337794 8320387477 0070444059 1560941044 7469927171 9851982871 1161845939 5264248069 7957875678 9732590612 1353375160 4978418071 9073947132 7275455053 0824277854 2239182932 6623301820 2694962171 0041384864 7891107636 2844707895 8474267263 8835353532 9559484653 6079421628 3168887378 2794293536 9562477925 9770585297 1769217978 2969388292 0191842562 6885438726 7466372272 6502170056 2146018680 3113171244 5585609601 8170733021 7058238537 0738451036 7982496401 4689893593 9512065248 1459062440 9628247796 8944250808 9323165521 1711214706 2272395319 6446950719 3514774003 9081085303 1441557927 7651961543 1649125434 9096543075 8964980555 3855171368 7627682794 2101488893 4535859995 4363118212 4502388915

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 ≡ 29 (mod 64) and therefore cannot be a square and 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 = 9006 significant digits (9000 digits displayed)

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

Input file is:  IO\33281741.cin
Certificate file is:  IO\33281741.chg
Found values of n, F and G.
    Number to be tested has 24506 digits.
    Modulus has 6883 digits.
Modulus is 28.08468837% 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 3859 digits.
        Done!  Time elapsed:  979000ms.
    Running CHG with h = 8, u = 3. Right endpoint has 3731 digits.
        Done!  Time elapsed:  1089985ms.
    Running CHG with h = 8, u = 3. Right endpoint has 3559 digits.
        Done!  Time elapsed:  1068875ms.
    Running CHG with h = 8, u = 3. Right endpoint has 3351 digits.
        Done!  Time elapsed:  3192875ms.
    Running CHG with h = 7, u = 2. Right endpoint has 3142 digits.
        Done!  Time elapsed:  288265ms.
    Running CHG with h = 7, u = 2. Right endpoint has 3035 digits.
        Done!  Time elapsed:  305860ms.
    Running CHG with h = 7, u = 2. Right endpoint has 2873 digits.
        Done!  Time elapsed:  272984ms.
    Running CHG with h = 6, u = 2. Right endpoint has 2654 digits.
        Done!  Time elapsed:  158063ms.
    Running CHG with h = 6, u = 2. Right endpoint has 2436 digits.
        Done!  Time elapsed:  104125ms.
    Running CHG with h = 7, u = 2. Right endpoint has 2217 digits.
        Done!  Time elapsed:  1019406ms.
    Running CHG with h = 5, u = 1. Right endpoint has 1694 digits.
        Done!  Time elapsed:  39609ms.
    Running CHG with h = 5, u = 1. Right endpoint has 1094 digits.
        Done!  Time elapsed:  54250ms.
    Running CHG with h = 5, u = 1. Right endpoint has 86 digits.
        Done!  Time elapsed:  276328ms.
A certificate has been saved to the file:  IO\33281741.chg

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

Testing a PRP called "IO\33281741.cin".

Pol[1, 1] with [h, u]=[5, 1] has ratio=3.085579488 E-1810 at X, ratio=5.25358162 E-1896 at Y, witness=3.
Pol[2, 1] with [h, u]=[4, 1] has ratio=0.947900942 at X, ratio=1.554372172 E-1008 at Y, witness=2.
Pol[3, 1] with [h, u]=[4, 1] has ratio=2.649698094 E-601 at X, ratio=7.92916436 E-601 at Y, witness=13.
Pol[4, 1] with [h, u]=[7, 2] has ratio=0.4466670850 at X, ratio=3.914050424 E-1046 at Y, witness=2.
Pol[5, 1] with [h, u]=[6, 2] has ratio=0.03200017180 at X, ratio=3.516061796 E-438 at Y, witness=5.
Pol[6, 1] with [h, u]=[6, 2] has ratio=0.681112975 at X, ratio=3.982040134 E-438 at Y, witness=2.
Pol[7, 1] with [h, u]=[6, 2] has ratio=0.1595103281 at X, ratio=3.946099173 E-438 at Y, witness=2.
Pol[8, 1] with [h, u]=[6, 2] has ratio=5.163606774 E-163 at X, ratio=6.28027568 E-324 at Y, witness=37.
Pol[9, 1] with [h, u]=[6, 2] has ratio=3.946069193 E-109 at X, ratio=3.620398834 E-216 at Y, witness=2.
Pol[10, 1] with [h, u]=[8, 3] has ratio=0.4865345660 at X, ratio=1.325794986 E-625 at Y, witness=11.
Pol[11, 1] with [h, u]=[8, 3] has ratio=0.3115881514 at X, ratio=9.34017439 E-626 at Y, witness=3.
Pol[12, 1] with [h, u]=[8, 3] has ratio=2.605300530 E-173 at X, ratio=1.395595990 E-515 at Y, witness=2.
Pol[13, 1] with [h, u]=[8, 3] has ratio=3.053858615 E-130 at X, ratio=8.74219750 E-387 at Y, witness=3.

Validated in 47 sec.


Congratulations! n is prime!

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