20090112, 00:34  #45 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
4277_{10} Posts 
2^(3*m)=(2^3)^m
(2^3)^m=8^m 8^m=(7+1)^m Okay, following it so far... (7+1)^m==(0+1)^m mod 7 How is the 7 eliminated so we know that 8^m==1^m? 1^m=1 so 2^(3*m)==1 mod 7 And the rest makes sense, so it's just that one part I'm confused on. 
20090112, 03:10  #46  
May 2007
Kansas; USA
23·467 Posts 
Quote:
Think of it like this: (7+1) == (1 mod 7) and (0+1) == (1 mod 7) therefore x represents the same integer 0<=x<=6 in both of the following: (7+1)^m == (x mod 7) and (0+1)^m == (x mod 7) therefore: (7+1)^m == (0+1)^m mod 7 therefore: 8^m == 1^m mod 7 Last fiddled with by gd_barnes on 20090112 at 03:11 

20090112, 06:22  #47  
"Jacob"
Sep 2006
Brussels, Belgium
11100011000_{2} Posts 
Quote:
If we compute (7+1)^m = 7^m+m*7^(m1)*1+ ... +m*7*1^(m1)+1^m all terms except 1^m contain a power of 7 greater than 0. So we can say that (7+1)^m is a multiple of 7 plus 1. More generally : (a*n+b)^m == b^m mod n. You could lookup some more explanation about modular arithmetics... Jacob 

20090113, 04:49  #48 
Feb 2006
Denmark
E6_{16} Posts 
From an unrelated PFGW search:
204912863*2^333332147 is composite: RES64: 69ED3420123456F2 (1.9872s+0.0003s) Not that much for a 2s test. 
20090430, 23:36  #49 
Apprentice Crank
Mar 2006
2×227 Posts 
I got 8 A's in one residue:
313*2^9195191 is not prime. LLR Res64: B9AA9AE1AAAD1AAD Time : 1317.195 sec. 
20090430, 23:43  #50 
Mar 2006
Germany
101110001000_{2} Posts 
other 8er's:
1199*2^1980961 is not prime. LLR Res64: CCBCCCF57770CCC0 1179*2^768931 is not prime. LLR Res64: 6C1D14411111CA41 1135*2^1391651 is not prime. LLR Res64: C656B5A966666D76 1165*2^1389731 is not prime. LLR Res64: F49A8666A66666F0 1179*2^1702641 is not prime. LLR Res64: A8A2C562AAAAAA6E the last one with 6 A's in a row! Last fiddled with by kar_bon on 20090430 at 23:45 
20090501, 17:05  #51 
May 2008
Wilmington, DE
101100100100_{2} Posts 
3311*2^1018941 CCCCCAC8484CF1AF 29.088 (Nice start)
10096*45^1125841 93663508540e5a65 4912.17 (Almost all numeric) 10096*45^361141 df8a177777509fac 230.17 (Craps anyone) 
20090603, 17:57  #52  
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
1761_{16} Posts 
Quote:
http://www.primenumbers.net/Renaud/eng/fermat1.html it explains why lots of numbers have the same residues time to repeat my question: does anyone know of a prp program that can display the full residue not just the RES64? i could then use it to find prps hopefully i expect the answer to be no but i would quite like a response 

20090603, 21:28  #53 
May 2005
2^{2}·11·37 Posts 
I've searched all my residues for the sequences of at least 6 digits / letters of the same kind:
Code:
1515*2^6183181 is not prime. LLR Res64: 5461D351111114AF > 6 x 1 25*2^6958671 is not prime. LLR Res64: 7D382AE72222229E > 6 x 2 59*2^16251321 is not prime. LLR Res64: E3CAA3333333171E > 7 x 3 736320585*2^7158451 is not prime. LLR Res64: 9604AF6744444483 > 6 x 4 2*3^1846021 is not prime. RES64: 214E3DFFA875B519. OLD64: B59B17A74A777777 > 6 x 7 736320585*2^933181 is not prime. LLR Res64: 18106436ABBBBBBA > 6 x B 
20090730, 23:49  #54 
A Sunny Moo
Aug 2007
USA (GMT5)
14151_{8} Posts 
Here's an interesting one that showed up today:
424*93^643371 is composite: RES64: [CAFFBBEEEA1063A4] (381.0392s+0.0157s) 
20090921, 03:50  #55 
A Sunny Moo
Aug 2007
USA (GMT5)
3×2,083 Posts 
Just now I was doing an offthewall search for MooMoo's "BEEF15BAD" residue with very small numbers (a fixed n search for k=31G, n=5, k*2^n+1), when I noticed a very weird pattern:
Code:
168195*2^5+1 = 5382241 is prime! (trial divisions) 168221*2^5+1 = 5383073 is prime! (trial divisions) 168225*2^5+1 = 5383201 is prime! (trial divisions) 168231*2^5+1 = 5383393 is prime! (trial divisions) 168239*2^5+1 = 5383649 is prime! (trial divisions) 168245*2^5+1 = 5383841 is prime! (trial divisions) 168251*2^5+1 = 5384033 is prime! (trial divisions) 168269*2^5+1 = 5384609 is prime! (trial divisions) 168281*2^5+1 = 5384993 is prime! (trial divisions) 168293*2^5+1 = 5385377 is prime! (trial divisions) 168305*2^5+1 = 5385761 is prime! (trial divisions) I'm sure there's a simple mathematical explanation for what I'm seeing here. You know, though I hate to sound "crankish"...if there is a simple mathematical proof that all k*2^5+1 are prime, then this could lead to a very simple way to find a 100 million digit prime that would qualify for the EFF prize! Heck on spending 3+ years per number searching 100 million digit numbers through GIMPS when you can just find one this way. (Of course, I'm sure there's something I'm missing that would preclude this, otherwise someone would have won the prize by now.) Edit: I'm seeing this on k*2^7+1 as well. Last fiddled with by mdettweiler on 20090921 at 03:56 
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