== ID:B2814010102EC =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 99999999999999999999999999899999999999999999999999999

Decimal size = 53
Binary size = 177

Started 11/02/2002 04:40:43 AM
Running time < 1s

Candidate certified prime

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

Proved prime with 'Primo 1.2.1'
by Patrick De Geest using a Pentium III 650 MHz chip.

[PRIMO - Primality Certificate]
Version=2.0.0 - beta 3
Format=2
ID=B2808009F5B3C
Created=10/21/2002 02:54:03 AM
TestCount=123
Status=Candidate certified prime

[Candidate]
File=C:\Program Files\Primo200\pwp989_757.in
Expression=(10^757-1) - 10^378
N$=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
HexadecimalSize=629
DecimalSize=757
BinarySize=2515

[Running Times]
Initialization=1.86s
1stPhase=36mn 38s
2ndPhase=4mn 42s
Total=41mn 22s

Proved prime with 'Primo 2.0.0 - beta 3'
by Patrick De Geest using a Pentium III 650 MHz chip.

(9)12468(9)1246 is prime, it is provable by a N+1 method.

C:\PrimeForm>pfgw -q10^2493-10^1246-1 -tp
PFGW Version 20010212.Win_Dev (Beta software, 'caveat utilitor')

Primality testing 10^2493-10^1246-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.94%
10^2493-10^1246-1 is prime! (9.780000 seconds)

(Timing using a Pentium III 650 Mhz chip).

(9)17988(9)1798 is prime, it is provable by a N+1 method.

C:\PrimeForm>pfgw -q10^3597-10^1798-1 -tp
PFGW Version 20010212.Win_Dev (Beta software, 'caveat utilitor')

Primality testing 10^3597-10^1798-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.93%
10^3597-10^1798-1 is prime! (24.220000 seconds)

(Timing using a Pentium III 650 Mhz chip).

(9)29178(9)2917 is prime, it is provable by a N+1 method.

C:\PrimeForm>pfgw -q10^5835-10^2917-1 -tp
PFGW Version 20010212.Win_Dev (Beta software, 'caveat utilitor')

Primality testing 10^5835-10^2917-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.94%
10^5835-10^2917-1 is prime! (56.910000 seconds)

(Timing using a Pentium III 650 Mhz chip).

The form of this prime is P = 10^(2n+1)-a*10^n-1 = 10^n(10^(n+1)-a)-1.
P+1 is "factorable" for 50%, so a N+1 methode is available
and PFGW (PrimeForm) can prove it.

Reference: http://groups.yahoo.com/group/primeform/message/2364

C:\PrimeForm>pfgw -q10^46069-10^23034-1 -tp
PFGW Version 20010212.Win_Dev (Beta software, 'caveat utilitor')

Primality testing 10^46069-10^23034-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.95%
10^46069-10^23034-1 is prime! (6449.950000 seconds)

(Timing using a Pentium III 650 Mhz chip).

Reference: http://groups.yahoo.com/group/primeform/message/3008

PrimeForm Output

C:\Pfgw4>pfgw -q"(10^95019-1)-10^47509" -tp
PFGW Version 1.2.0 for Windows [FFT v23.8]

Primality testing (10^95019-1)-10^47509 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Running N+1 test using discriminant 7, base 2+sqrt(7)
Running N+1 test using discriminant 7, base 7+sqrt(7)
Running N+1 test using discriminant 7, base 8+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.95%
(10^95019-1)-10^47509 is prime! (8769.3868s+0.0064s)

Reference: http://groups.yahoo.com/group/primeform/message/3033

PrimeForm Output

C:\Pfgw4>pfgw -q"(10^104281-1)-10^52140" -tp
PFGW Version 1.2.0 for Windows [FFT v23.8]

Primality testing (10^104281-1)-10^52140 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.95%
(10^104281-1)-10^52140 is prime! (3130.3335s+0.0077s)

By Darren Bedwell

http://zerosink.blogspot.com/2010/11/new.html

Primality testing 10^(2*67404+1)-1*10^67404-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.95%
10^(2*67404+1)-1*10^67404-1 is prime! (4490.0911s+0.0381s)

By Ryan Propper and Serge Batalov

http://primes.utm.edu/primes/page.php?id=132851

Command: /home/caldwell/clientpool/1/pfgw64 -tp -q"10^1888529-10^944264-1" 2>&1
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 10^1888529-10^944264-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 34.95%
10^1888529-10^944264-1 is prime! (391118.1462s+0.0915s)
[Elapsed time: 4.53 days]