7*(10^3-1)/9 - 4*(10^2+1)
*** VFYPR 1.13F F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0
N=(37*10^3-73)/99
N=373
*** N is prime!
Time: 0 sec
7*(10^15-1)/9 - 4*(10^14+1)
== ID:B286400BBC272 =============================================
PRIMO 1.2.1 - Primality Certificate
-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 377777777777773
Decimal size = 15
Binary size = 49
-----------------------------------------------------------------
1) EC Test
-----------------------------------------------------------------
N = Candidate
S = 7861
R = 48057211417
A = 0
B = 2
T = 1
-----------------------------------------------------------------
2) SPP Test
-----------------------------------------------------------------
N = R of preceding test
Started 01/21/2003 03:25:05 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.
7*(10^55-1)/9 - 4*(10^54+1)
== ID:B286804901376 =============================================
PRIMO 1.2.1 - Primality Certificate
-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 3777777777777777777777777777777777777777777777777777773
Decimal size = 55
Binary size = 182
Started 01/25/2003 09:15:51 PM
Running time < 1s
Candidate certified prime
=================================================================
Proved prime with 'Primo 1.2.1'
by Patrick De Geest using a Pentium III 650 MHz chip.
7*(10^69-1)/9 - 4*(10^68+1)
== ID:B2869006994E8 =============================================
PRIMO 1.2.1 - Primality Certificate
-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 377777777777777777777777777777777777777777777777777777777777\
777777773
Decimal size = 69
Binary size = 228
Started 01/26/2003 01:55:19 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.
7*(10^85-1)/9 - 4*(10^84+1)
== ID:B2869007C5BBE =============================================
PRIMO 1.2.1 - Primality Certificate
-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 377777777777777777777777777777777777777777777777777777777777\
7777777777777777777777773
Decimal size = 85
Binary size = 281
Started 01/26/2003 02:15:49 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.
7*(10^87-1)/9 - 4*(10^86+1)
== ID:B2869007E79B2 =============================================
PRIMO 1.2.1 - Primality Certificate
-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 377777777777777777777777777777777777777777777777777777777777\
777777777777777777777777773
Decimal size = 87
Binary size = 288
Started 01/26/2003 02:18:08 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.
7*(10^157-1)/9 - 4*(10^156+1)
== ID:B286903F937BC =============================================
PRIMO 1.2.1 - Primality Certificate
-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 377777777777777777777777777777777777777777777777777777777777\
777777777777777777777777777777777777777777777777777777777777\
7777777777777777777777777777777777773
Decimal size = 157
Binary size = 521
Started 01/26/2003 06:31:04 PM
Running time 2s
Candidate certified prime
=================================================================
Proved prime with 'Primo 1.2.1'
by Patrick De Geest using a Pentium III 650 MHz chip.
7*(10^10747-1)/9 - 4*(10^10746+1)
3-PRP!
7*(10^15769-1)/9 - 4*(10^15768+1)
by Greg Childers (email1) (email2) February 28, 2006
The complete proof of (34*10^15768-43)/9 with both
the Primo and CHG certificates is posted at
http://www.pa.uky.edu/~childers/certs/P15769.zip
This is a fine proof, combining state-of-the-art factorization
with three types of primality testing and proving
(BLS, CHG, ECPP). David Broadhurst added this prime also in
the "Caldwell-illegitimate" appendix located at
http://groups.yahoo.com/group/primeform/files/NTG/gigantic.txt
All the digits are prime digits as well.
Therefore this number is included in WONplate 150.
From the README.TXT file :
Proof of the primality of N=(34*10^15768-43)/9
1. The enclosed factors.txt contains prime factors of N+1, and
log(product of factors)/log(N) = 0.27. All factors are relatively small
and easy to prove with the exception of the 2454-digit prime
Phi(7884,10)/(3951358309*43519681*11668781326071061*50594869824289387600141*
698321409620914728282889*10226827901261393154083521*
2768214255974519676483980192126941)
This was proven prime using Primo, and the certificate is in the enclosed
file Phi7884_10.out
2. The BLS tests were completed by PFGW using these prime factors as
a helper file.
3. The proof was then completed using the CHG script of John Renze.
The precision was adjusted to \p9000 to avoid errors. The CHG certificate
is in the enclosed CHG_P15769.out.
4. A verification of this CHG certificate was performed by David
Broadhurst's verifier, chgcertd.gp. The output of the verification
is in the enclosed CHG_P15769_ver.out.
Special thanks to Paul Underwood for proving the primality of the 2454-
digit prime factor using Primo, John Renze and David Broadhurst for
creating the CHG script and verifier, and to all the authors of PFGW,
Primo, and Pari/GP.
Greg Childers
The Yahoo discussion Primeform Group has been discontinued.
Here is a reconstruction in html of the messages from February 2006.
Message 6988 Message 6989 Message 6990 Message 6991 Message 6992 Message 6993
Message 6994 Message 6995 Message 6998 Message 7003 Message 7006 Message 7024 Message 7033
7*(10^31317-1)/9 - 4*(10^31319+1)
3,5,7-PRP!
PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8].
7*(10^40959-1)/9 - 4*(10^40958+1)
3,5,7-PRP!
PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8].
7*(10^45805-1)/9 - 4*(10^45804+1)
3,5,7-PRP!
PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8].
7*(10^46567-1)/9 - 4*(10^46566+1)
3,5,7-PRP!
PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8].
7*(10^51009-1)/9 - 4*(10^51008+1)
PFGW Version 3.3.6.20100908.Win_Stable [GWNUM 25.14]
Primality testing (34*10^51008-43)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 3+sqrt(11)
(34*10^51008-43)/9 is Fermat and Lucas PRP! (1752.3648s+0.0031s)
7*(10^80163-1)/9 - 4*(10^80162+1)
PFGW Version 3.4.2.64BIT.20101019.Win_Dev [GWNUM 26.4]
Primality testing (34*10^80162-43)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 5, base 4+sqrt(5)
Calling N+1 BLS with factored part 0.03% and helper 0.01% (0.09% proof)
(34*10^80162-43)/9 is Fermat and Lucas PRP! (2210.3065s+0.0029s)
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( © All rights reserved ) - Last modified : March 15, 2023.
Patrick De Geest - Belgium 
- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com