```

(32*10^5-23)/99

*** VFYPR 1.13F  F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0
N=(32*10^5-23)/99
N=32323
*** N is prime!
Time: 0 sec

(32*10^9-23)/99

*** VFYPR 1.13F  F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0
N=(32*10^9-23)/99
N=323232323
*** N is prime!
Time: 0 sec

(32*10^11-23)/99

*** VFYPR 1.13F  F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0
N=(32*10^11-23)/99
N=32323232323
Factor: 2 divides N - 1
Factor: 2^2 divides N + 1
Factor: 3^2 divides N - 1
Factor: 29 divides N - 1
Factor: 53 divides N + 1
Factor: 83 divides N - 1
Factorization results: F1=0.4412 F2=0.2214
F1=43326
F2=212
Pass: gcd(3^((N-1)/2) - 1, N) = 1: R20=32323232321
Pass: 3^(N-1) = 1 (mod N): R20=1
Pass: gcd(U{(N+1)/2}, N) = 1: d=5 p=1 q=-1 R20=16198539604
Pass: U{N+1} = 0 (mod N): d=5 p=1 q=-1 R20=0
Pass: gcd(3^((N-1)/3) - 1, N) = 1: R20=21870921832
Pass: gcd(3^((N-1)/29) - 1, N) = 1: R20=29159760567
Pass: gcd(U{(N+1)/53}, N) = 1: d=5 p=1 q=-1 R20=28402242962
Pass: gcd(3^((N-1)/83) - 1, N) = 1: R20=23942302048
BLS tests passed: F1=0.4412 F2=0.2214
Main divisor test: F1=0.4126 F2=0.2214 G=0.6339 S=0.0000 T=1
G=4592556
Main divisor test passed: 1/1
*** N is prime!
Time: 0 sec

(32*10^3015-23)/99

== ID:B265D044D8EA2 =============================================

PRIMO 0.1.0 - Primality Certificate

Started 08.20.2001 08:03:11 PM
Running time 290h 42mn 5s

Candidate certified prime

=================================================================

Proved prime with 'Primo' by Hans Rosenthal.
The zipped file "323_1507.zip" is 913 KB.
When unpacked the file "Primo-B265D044D8EA2-001.out" is 2092 KB
and is available on demand by simple email request.

(32*10^3407-23)/99

== ID:B268C0053A5D4 =============================================

PRIMO 1.0.0 - Primality Certificate

Started 10.06.2001 01:31:51 AM
Running time 273h 17mn 24s

Candidate certified prime

=================================================================

Proved prime with 'Primo' by Hans Rosenthal.
The zipped file "323_1703.zip" is 1181 KB.
When unpacked the file "Primo-B268C0053A5D4-S.out" is 2690 KB
and is available on demand by simple email request.

(32*10^6959-23)/99

Hans Rosenthal, July 13, 2003
announced via a message in Number Theory List (NMBRTHRY@LISTSERV.NODAK.EDU)
that this SUPP is prime !
He thereby also established a new Primo ECPP world record
performed on a single monoprocessor computersystem.

[PRIMO - Task Report]
Version=2.0.0 - beta 4
ID=B290404F2E14A
Created=06.30.2003 11:03:46 PM

[Common]
Path=C:\Programme\Primo\Work\
Selected=1
Processed=1
Certified=1
Candidate #1=Certified, 5459h 25mn 0s

[Candidate #1]
Input=Primo-B276304567D8C-001.tmp Output=Primo-B276304567D8C-001.out
Status=Candidate certified prime

Proved prime with 'Primo 2.0.0 - beta 4'
by Hans Rosenthal using an Athlon 1.4 GHz. Running time
amounts to approximately 527 days (18 months). He thereby
established a new current ECPP (Elliptic Curve Primality
Proving) record !

Primality certificate available by clicking the following link :
http://www.ellipsa.net/primo/ecpp6959.zip
A diary written by Hans Rosenthal can be consulted at
http://www.ellipsa.net/primo/ecpp6959_diary.txt

The verification of the certificate was done with the same
Primo 2.0.0 beta 4 on the same Athlon 1.4 GHz and took
50.6 hours.

[PRIMO - Task Report]
Version=2.0.0 - beta 4
ID=B290C04024258
Created=07.08.2003 06:40:56 PM

[Common]
Path=C:\Programme\Primo\Work\
Selected=1
Processed=1
Valid=1
Certificate #1=Valid, 50h 38mn 36s

[Certificate #1]
Output=ecpp6959.out
Status=Valid certificate

(32*10^9599-23)/99

PFGW 1.1 test for probable primality in basis
3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,
and 251

(32*10^9599-23)/99 is 3-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 5-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 7-PRP! (19.170000 seconds)
(32*10^9599-23)/99 is 11-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 13-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 17-PRP! (19.120000 seconds)
(32*10^9599-23)/99 is 19-PRP! (19.170000 seconds)
(32*10^9599-23)/99 is 23-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 29-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 31-PRP! (19.120000 seconds)
(32*10^9599-23)/99 is 37-PRP! (19.170000 seconds)
(32*10^9599-23)/99 is 41-PRP! (19.170000 seconds)
(32*10^9599-23)/99 is 43-PRP! (19.120000 seconds)
(32*10^9599-23)/99 is 47-PRP! (19.110000 seconds)
(32*10^9599-23)/99 is 53-PRP! (19.170000 seconds)
(32*10^9599-23)/99 is 59-PRP! (19.120000 seconds)
(32*10^9599-23)/99 is 61-PRP! (19.120000 seconds)
(32*10^9599-23)/99 is 251-PRP! (19.110000 seconds)

From: Hans Rosenthal
Sent: Sun 21/10/12 03:19
Subject: SUPP (32*10^9599-23)/99 is prime

Dear Patrick,

After 293 days (32*10^9599-23)/99 has now been certified prime with Primo.

Best regards

Hans

(32*10^11399-23)/99

PFGW 1.1 test for probable primality in basis
3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,
and 251

(32*10^11399-23)/99 is 3-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 5-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 7-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 11-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 13-PRP! (30.320000 seconds)
(32*10^11399-23)/99 is 17-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 19-PRP! (30.320000 seconds)
(32*10^11399-23)/99 is 23-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 29-PRP! (30.380000 seconds)
(32*10^11399-23)/99 is 31-PRP! (30.380000 seconds)
(32*10^11399-23)/99 is 37-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 41-PRP! (30.420000 seconds)
(32*10^11399-23)/99 is 43-PRP! (30.380000 seconds)
(32*10^11399-23)/99 is 47-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 53-PRP! (30.370000 seconds)
(32*10^11399-23)/99 is 59-PRP! (30.430000 seconds)
(32*10^11399-23)/99 is 61-PRP! (30.380000 seconds)
(32*10^11399-23)/99 is 251-PRP! (30.320000 seconds)

(32*10^16593-23)/99

PFGW 1.1 test for probable primality in basis
3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,
and 251

(32*10^16593-23)/99 is 3-PRP! (76.240000 seconds)
(32*10^16593-23)/99 is 5-PRP! (75.080000 seconds)
(32*10^16593-23)/99 is 7-PRP! (75.190000 seconds)
(32*10^16593-23)/99 is 11-PRP! (76.900000 seconds)
(32*10^16593-23)/99 is 13-PRP! (74.870000 seconds)
(32*10^16593-23)/99 is 17-PRP! (76.510000 seconds)
(32*10^16593-23)/99 is 19-PRP! (74.870000 seconds)
(32*10^16593-23)/99 is 23-PRP! (75.470000 seconds)
(32*10^16593-23)/99 is 29-PRP! (76.730000 seconds)
(32*10^16593-23)/99 is 31-PRP! (74.860000 seconds)
(32*10^16593-23)/99 is 37-PRP! (75.790000 seconds)
(32*10^16593-23)/99 is 41-PRP! (74.970000 seconds)
(32*10^16593-23)/99 is 43-PRP! (74.800000 seconds)
(32*10^16593-23)/99 is 47-PRP! (75.800000 seconds)
(32*10^16593-23)/99 is 53-PRP! (77.340000 seconds)
(32*10^16593-23)/99 is 59-PRP! (76.570000 seconds)
(32*10^16593-23)/99 is 61-PRP! (76.020000 seconds)
(32*10^16593-23)/99 is 251-PRP! (75.030000 seconds)

(32*10^25883-23)/99

PFGW 1.1 test for probable primality in bases
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, and 251

(32*10^25883-23)/99 is 3-PRP! (299.180000 seconds)
(32*10^25883-23)/99 is 5-PRP! (296.650000 seconds)
(32*10^25883-23)/99 is 7-PRP! (298.020000 seconds)
(32*10^25883-23)/99 is 11-PRP! (297.860000 seconds)
(32*10^25883-23)/99 is 13-PRP! (295.550000 seconds)
(32*10^25883-23)/99 is 17-PRP! (296.650000 seconds)
(32*10^25883-23)/99 is 19-PRP! (296.430000 seconds)
(32*10^25883-23)/99 is 23-PRP! (297.590000 seconds)
(32*10^25883-23)/99 is 29-PRP! (297.200000 seconds)
(32*10^25883-23)/99 is 31-PRP! (297.310000 seconds)
(32*10^25883-23)/99 is 37-PRP! (296.870000 seconds)
(32*10^25883-23)/99 is 41-PRP! (295.880000 seconds)
(32*10^25883-23)/99 is 43-PRP! (296.220000 seconds)
(32*10^25883-23)/99 is 47-PRP! (296.380000 seconds)
(32*10^25883-23)/99 is 53-PRP! (296.320000 seconds)
(32*10^25883-23)/99 is 59-PRP! (298.360000 seconds)
(32*10^25883-23)/99 is 61-PRP! (296.330000 seconds)
(32*10^25883-23)/99 is 67-PRP! (296.270000 seconds)
(32*10^25883-23)/99 is 71-PRP! (300.010000 seconds)
(32*10^25883-23)/99 is 73-PRP! (296.320000 seconds)
(32*10^25883-23)/99 is 79-PRP! (297.040000 seconds)
(32*10^25883-23)/99 is 83-PRP! (297.640000 seconds)
(32*10^25883-23)/99 is 89-PRP! (295.830000 seconds)
(32*10^25883-23)/99 is 97-PRP! (298.140000 seconds)
(32*10^25883-23)/99 is 101-PRP! (300.060000 seconds)
(32*10^25883-23)/99 is 103-PRP! (297.480000 seconds)
(32*10^25883-23)/99 is 107-PRP! (297.090000 seconds)
(32*10^25883-23)/99 is 109-PRP! (302.150000 seconds)
(32*10^25883-23)/99 is 113-PRP! (297.360000 seconds)
(32*10^25883-23)/99 is 127-PRP! (297.090000 seconds)
(32*10^25883-23)/99 is 251-PRP! (300.170000 seconds)

```