# Primality Proof of phi(24691,92365)

## OpenPFGW

```Primality testing (92365^24691-1)/92364 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 7
Calling Brillhart-Lehmer-Selfridge with factored part 26.87%
(92365^24691-1)/92364 is PRP! (431.0973s+0.0012s)
```

## CHG

```                      GP/PARI CALCULATOR Version 2.13.3 (released)
amd64 running linux (x86-64/GMP-6.2.1 kernel) 64-bit version
compiled: Oct 25 2021, gcc version 11.2.0 (Ubuntu 11.2.0-10ubuntu1)
(readline v8.1 enabled, extended help enabled)

Copyright (C) 2000-2020 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT
ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000, nbthreads = 8
***   Warning: new stack size = 536870912 (512.000 Mbytes).
realprecision = 50014 significant digits (50000 digits displayed)

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

Input file is:  92365_24691.in
Certificate file is:  92365_24691.out
Found values of n, F and G.
Number to be tested has 122599 digits.
Modulus has 32940 digits.
Modulus is 26.867420021110790498% 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 = 14, u = 6. Right endpoint has 23782 digits.
Done!  Time elapsed:  9933860ms.
Running CHG with h = 14, u = 6. Right endpoint has 23384 digits.
Done!  Time elapsed:  9669012ms.
Running CHG with h = 14, u = 6. Right endpoint has 22961 digits.
Done!  Time elapsed:  8906685ms.
Running CHG with h = 13, u = 5. Right endpoint has 22526 digits.
Done!  Time elapsed:  4711596ms.
Running CHG with h = 13, u = 5. Right endpoint has 22312 digits.
Done!  Time elapsed:  4879146ms.
Running CHG with h = 13, u = 5. Right endpoint has 21986 digits.
Done!  Time elapsed:  4620042ms.
Running CHG with h = 13, u = 5. Right endpoint has 21690 digits.
Done!  Time elapsed:  4811585ms.
Running CHG with h = 13, u = 5. Right endpoint has 21304 digits.
Done!  Time elapsed:  4713306ms.
Running CHG with h = 13, u = 5. Right endpoint has 20880 digits.
Done!  Time elapsed:  5129083ms.
Running CHG with h = 13, u = 5. Right endpoint has 20321 digits.
Done!  Time elapsed:  6330445ms.
Running CHG with h = 11, u = 4. Right endpoint has 19545 digits.
Done!  Time elapsed:  1957517ms.
Running CHG with h = 11, u = 4. Right endpoint has 19173 digits.
Done!  Time elapsed:  1912417ms.
Running CHG with h = 11, u = 4. Right endpoint has 18693 digits.
Done!  Time elapsed:  1777571ms.
Running CHG with h = 11, u = 4. Right endpoint has 18162 digits.
Done!  Time elapsed:  1921192ms.
Running CHG with h = 11, u = 4. Right endpoint has 17539 digits.
Done!  Time elapsed:  2069536ms.
Running CHG with h = 9, u = 3. Right endpoint has 16655 digits.
Done!  Time elapsed:  516441ms.
Running CHG with h = 9, u = 3. Right endpoint has 16219 digits.
Done!  Time elapsed:  583214ms.
Running CHG with h = 9, u = 3. Right endpoint has 15931 digits.
Done!  Time elapsed:  608317ms.
Running CHG with h = 9, u = 3. Right endpoint has 15546 digits.
Done!  Time elapsed:  515062ms.
Running CHG with h = 9, u = 3. Right endpoint has 14741 digits.
Done!  Time elapsed:  563800ms.
Running CHG with h = 9, u = 3. Right endpoint has 13760 digits.
Done!  Time elapsed:  531461ms.
Running CHG with h = 7, u = 2. Right endpoint has 12417 digits.
Done!  Time elapsed:  109425ms.
Running CHG with h = 7, u = 2. Right endpoint has 11186 digits.
Done!  Time elapsed:  128285ms.
Running CHG with h = 7, u = 2. Right endpoint has 9563 digits.
Done!  Time elapsed:  111335ms.
Running CHG with h = 5, u = 1. Right endpoint has 7077 digits.
Done!  Time elapsed:  11673ms.
Running CHG with h = 5, u = 1. Right endpoint has 2954 digits.
Done!  Time elapsed:  9736ms.
A certificate has been saved to the file:  92365_24691.out

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

Testing a PRP called "92365_24691.in".

Pol[1, 1] with [h, u]=[5, 1] has ratio=2.6465957398628042790 E-4332 at X, ratio=6.802214999566733530 E-6746 at Y, witness=2.
Pol[2, 1] with [h, u]=[4, 1] has ratio=1.1078032544378937300 E-3237 at X, ratio=5.007041483512086564 E-4123 at Y, witness=2.
Pol[3, 1] with [h, u]=[7, 2] has ratio=1.6582553525473062566 E-8245 at X, ratio=2.491391136783331361 E-4974 at Y, witness=2.
Pol[4, 1] with [h, u]=[7, 2] has ratio=8.371439846877536977 E-1624 at X, ratio=7.008100510988899974 E-3247 at Y, witness=2.
Pol[5, 1] with [h, u]=[7, 2] has ratio=5.250045138392966626 E-2419 at X, ratio=2.4900086218722161226 E-2462 at Y, witness=2.
Pol[6, 1] with [h, u]=[9, 3] has ratio=1.4343076394692276521 E-4029 at X, ratio=1.4343076394692276521 E-4029 at Y, witness=2.
Pol[7, 1] with [h, u]=[9, 3] has ratio=4.002284472103392102 E-1371 at X, ratio=1.6829312818585009175 E-2944 at Y, witness=2.
Pol[8, 1] with [h, u]=[9, 3] has ratio=0.8015436887843473920 at X, ratio=2.0721720076172118000 E-2415 at Y, witness=2.
Pol[9, 1] with [h, u]=[9, 3] has ratio=0.6073594906062747239 at X, ratio=1.2861160173364039898 E-1154 at Y, witness=2.
Pol[10, 1] with [h, u]=[9, 3] has ratio=0.07773694167387068868 at X, ratio=3.711868172410508133 E-866 at Y, witness=17.
Pol[11, 1] with [h, u]=[9, 3] has ratio=8.124493168651349852 E-351 at X, ratio=2.370278033714354033 E-1308 at Y, witness=2.
Pol[12, 1] with [h, u]=[11, 4] has ratio=8.726053724922577115 E-1563 at X, ratio=1.2544068202618376844 E-3533 at Y, witness=2.
Pol[13, 1] with [h, u]=[11, 4] has ratio=4.727090619190449929 E-626 at X, ratio=2.5087502389258144796 E-2496 at Y, witness=17.
Pol[14, 1] with [h, u]=[11, 4] has ratio=0.8383848989493183355 at X, ratio=3.561555659258786197 E-2122 at Y, witness=5.
Pol[15, 1] with [h, u]=[11, 4] has ratio=5.781347762738193315 E-1019 at X, ratio=2.700898703368168856 E-1923 at Y, witness=2.
Pol[16, 1] with [h, u]=[11, 4] has ratio=0.12338652648716530740 at X, ratio=2.6457980489074973737 E-1487 at Y, witness=2.
Pol[17, 1] with [h, u]=[13, 5] has ratio=2.631323782125048675 E-1072 at X, ratio=5.065093821607389414 E-3881 at Y, witness=2.
Pol[18, 1] with [h, u]=[13, 5] has ratio=1.991009881036662560 E-559 at X, ratio=7.066042726336390891 E-2794 at Y, witness=3.
Pol[19, 1] with [h, u]=[13, 5] has ratio=1.1761336189886154868 E-1273 at X, ratio=2.823326515990305194 E-2122 at Y, witness=2.
Pol[20, 1] with [h, u]=[13, 5] has ratio=1.0227905580481556009 E-386 at X, ratio=1.5823416859980017464 E-1929 at Y, witness=11.
Pol[21, 1] with [h, u]=[13, 5] has ratio=0.6013777104567411717 at X, ratio=3.219714014154501060 E-1483 at Y, witness=5.
Pol[22, 1] with [h, u]=[13, 5] has ratio=2.6779252617103771903 E-813 at X, ratio=5.008269833469465246 E-1629 at Y, witness=17.
Pol[23, 1] with [h, u]=[12, 5] has ratio=6.403660102627984046 E-238 at X, ratio=1.4651746356935392232 E-1070 at Y, witness=2.
Pol[24, 1] with [h, u]=[14, 6] has ratio=2.6432755870051564283 E-1163 at X, ratio=7.755291068816337700 E-2609 at Y, witness=2.
Pol[25, 1] with [h, u]=[14, 6] has ratio=2.0671158861869316187 E-163 at X, ratio=2.0262683760754481307 E-2539 at Y, witness=2.
Pol[26, 1] with [h, u]=[14, 6] has ratio=3.614123592378731148 E-399 at X, ratio=1.1137610238961956156 E-2390 at Y, witness=3.

Validated in 19 sec.

Congratulations! n is prime!
Goodbye!
```
A copy of the CHG certificate `92365_24691.out` (3.2MB) is included in: `92365_24691.zip`.

## Helper File

Based on factorization of N-1 and N+1:
```Phi(12345,92365)/382141463174433542855941
2548497703465213832097675992723
223201688124515521307049781
382141463174433542855941
349006742447838757179631
767470365893753920201
20852837822718331
5338140371167861
89640839069531
3263503041661
1016318173501
513608066053
8531200861
8468620621
2843795197
103920211
45500431
25513001
10468561
811931
233371
46183
24691
20101
8231
4651
331
31
29
13
11
7
7
5
3
2
```

## Prime Factor Certification

A Primo-format certificate for `Phi(12345,92365)/382141463174433542855941` (32630 digits) generated by CM-ECPP is included in: `92365_24691.zip`

Tom Wu