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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*(10^2767-1)/9 - 4*(10^2766+1) [PRIMO - Primality Certificate] Version=2.1.1 WebSite=http://www.ellipsa.net/ Format=3 ID=B290F008096B3 Created=07/11/2003 02:20:53 AM TestCount=408 Status=Candidate certified prime [Running Times] Initialization=26.74s 1stPhase=147h 36mn 9s 2ndPhase=7h 39mn 12s Total=155h 15mn 48s [Candidate] File=C:\Primo IN & OUT files\PDP2767.in Expression=7*(10^2767-1)/9-4*(10^2766+1) N$=52BF6CFE1941AED6FF27CC641A5E53DC6D2F6D2C61085AC9D42411041F1CE6BEC35CA8626D573E... HexadecimalSize=2298 DecimalSize=2767 BinarySize=9191 Proved prime with 'Primo 2.1.1' by Patrick De Geest using a 3000 MHz Pentium 4 cpu. Certificate Primo-B290F008096B3-01.out available by simple email request (897 KB).
7*(10^3381-1)/9 - 4*(10^3380+1) [PRIMO - Primality Certificate] Version=2.2.0 beta 1 WebSite=http://www.ellipsa.net/ Format=3 ID=B295803B24992 Created=09/22/2003 05:14:20 PM TestCount=483 Status=Candidate certified prime [Running Times] Initialization=44.11s 1stPhase=216h 24mn 11s 2ndPhase=36h 35mn 30s Total=253h 0mn 25s [Candidate] File=C:\Primo IN & OUT files\PDP3381.in Expression=7*(10^3381-1)/9-4*(10^3380+1) N$=418C816ABF7613C6599C318BA21FB104363F4EAE39C859199FC161D2B8172E0A9C7D917F8CE12F... HexadecimalSize=2808 DecimalSize=3381 BinarySize=11231 --------------------------------------------------------- --------------------------------------------------------- Proved prime with 'Primo 2.2.0 - beta 1' by Patrick De Geest using a 3000 MHz Pentium 4 cpu. Certificate Primo-B295803B24992-01.out available by simple email request (1279 KB).
7*(10^3877-1)/9 - 4*(10^3876+1) Primo certificate by courtesy of Ray Chandler (Feb 7, 2013) from factrodb.com http://factordb.com/cert.php?id=1000000000010853900 [PRIMO - Primality Certificate] Version=4.0.1 - LX64 WebSite=http://www.ellipsa.eu/ Format=3 ID=B36B7048E1F3D Created=02-04-2013 09:13:50 pm TestCount=494 Status=Candidate certified prime [Comments] Put here any comment... [Running Times] Initialization=7.10s 1stPhase=27094s 2ndPhase=8183s Total=35284s [Candidate] File=/home/ray/Documents/primo/fdb_knrr/pdp-3877-37w3/primo_1000000000010853900.in N$=346040B0903C1F24C7FC495CCCA676CBEC4072EF151051544A0B6E00C840212BB45626DEE3CAD5... HexadecimalSize=3220 DecimalSize=3877 BinarySize=12878
7*(10^5209-1)/9 - 4*(10^5208+1) CHG proof by courtesy of Ray Chandler (Feb 7, 2013) http://mada.la.coocan.jp/nrr/cert/3/37773_5208.zip Attributed to Serge Batalov around 12/15/2010.
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

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)