# Primality Proof of phi(9283,32556)

## OpenPFGW

```Primality testing Phi(9283,32556) [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 29.24%
Phi(9283,32556) is PRP! (622.1309s+0.0049s)
```

## CHG

```   realprecision = 15008 significant digits (15000 digits displayed)

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

Input file is:  32556_9283.in
Certificate file is:  32556_9283.out
Found values of n, F and G.
Number to be tested has 41887 digits.
Modulus has 12249 digits.
Modulus is 29.242700561227281416% 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 = 6, u = 2. Right endpoint has 5141 digits.
Done!  Time elapsed:  619864ms.
Running CHG with h = 6, u = 2. Right endpoint has 4551 digits.
Done!  Time elapsed:  309118ms.
Running CHG with h = 5, u = 1. Right endpoint has 3666 digits.
Done!  Time elapsed:  16660ms.
Running CHG with h = 5, u = 1. Right endpoint has 3248 digits.
Done!  Time elapsed:  14797ms.
Running CHG with h = 5, u = 1. Right endpoint has 2413 digits.
Done!  Time elapsed:  17381ms.
Running CHG with h = 5, u = 1. Right endpoint has 742 digits.
Done!  Time elapsed:  254456ms.
A certificate has been saved to the file:  32556_9283.out

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

Testing a PRP called "32556_9283.in".

Pol[1, 1] with [h, u]=[5, 1] has ratio=1.6391805052503673792 E-2389 at X, ratio=3.0487401337550959476 E-3130 at Y, witness=3.
Pol[2, 1] with [h, u]=[4, 1] has ratio=2.274605962029519074 E-1671 at X, ratio=2.315687251425674674 E-1671 at Y, witness=2.
Pol[3, 1] with [h, u]=[4, 1] has ratio=2.491325468744583574 E-836 at X, ratio=5.819011595899484830 E-836 at Y, witness=43.
Pol[4, 1] with [h, u]=[4, 1] has ratio=1.0107237831606918645 E-419 at X, ratio=2.6538923000009495060 E-418 at Y, witness=11.
Pol[5, 1] with [h, u]=[6, 2] has ratio=1.5879682798710421479 E-223 at X, ratio=2.222854004357033572 E-1771 at Y, witness=2.
Pol[6, 1] with [h, u]=[6, 2] has ratio=1.8824321194576357990 E-591 at X, ratio=4.317527851792719132 E-1181 at Y, witness=5.

Validated in 1 sec.

Congratulations! n is prime!
Goodbye!
```
A copy of the CHG certificate `32556_9283.out` (460KB) is included in: `32556_9283.zip`.

## Helper File

Based on factorization of N-1 and N+1:
```Phi(4641,32556)/(9283*27847*84419168107)
Phi(238,32556)/5945479
Phi(91,32556)/(5279*114661*241333)
3310094458802436352022645282885682308639377717432718293966430431410299075627222538078257300468629
166364075933804606897730632371159261288294481657345170415330517544747213227503929264551607073803
702228716743738506336271684406175094625370173577358222919827
2243718128031182700204349955662838879676117433027
300131567247657073728143777923625332167810110461
16074579753223402966060731951806935867833825343
6942257334722550169282410210148391653103
60333006542335543084646750464972555903
138083697584507166833199009903639941
686334289568597756872063
509060450112300546761183
141268317019711855645699
28187836949900053616569
8055536583557978132489
198984956787013041493
14710832227596782821
7457975774881479121
4901064650867507893
1952847144377897899
379378626376299811
315580101057298663
96887763471478099
78369203765408011
24332619775975759
5025858498869473
10363792922533
8357562203131
5568268663777
3418648563373
101454273397
90731425459
84419168107
6531879511
308625787
233779547
140220523
34704391
27114389
7239961
5945479
4723321
2629927
1541051
1379449
1168337
407083
389743
245039
241333
114661
74257
63649
30941
29989
27847
9283
5851
5279
4999
4651
4597
3571
2713
2687
2377
2347
2339
1327
911
859
613
547
443
313
239
157
137
131
103
97
79
43
31
17
13
13
7
7
3
2
2
```

## Prime Factor Certification

Signed Primo certificate for `Phi(4641,32556)/(9283*27847*84419168107)` (10378 digits): `ecpp10378.zip`

Tom Wu