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 2009-12-08, 14:43 #45 Mini-Geek Account Deleted     "Tim Sorbera" Aug 2006 San Antonio, TX USA 10000101010112 Posts Riesel Base 707 Conjectured k = 14 Found Primes: Code: 2*707^350-1 4*707^3-1 6*707^1-1 8*707^4-1 10*707^1-1 Remaining k's: Tested to n=2500 12*707^n-1 Base Released
 2009-12-08, 14:50 #46 Mini-Geek Account Deleted     "Tim Sorbera" Aug 2006 San Antonio, TX USA 10000101010112 Posts Riesel Base 713 Riesel Base 713 Conjectured k = 8 Found Primes: Code: 2*713^2-1 4*713^1-1 6*713^2-1 Conjecture Proven
2009-12-08, 16:05   #47
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

101110100012 Posts

Quote:
 Originally Posted by Mini-Geek Sierp Base 1000 Conjectured k = 12 Found Primes: Code: 3*1000^1+1 4*1000^1+1 6*1000^3+1 7*1000^1+1 9*1000^1+1 Remaining k's: Tested to n=10K 10*1000^n+1 Trivial Factor Eliminations: 2 5 8 11 GFN Eliminations: 1 Base Released (If k=10 can be eliminated for some algebraic/trivial reason, I don't see why. The automatic PFGW script didn't eliminate it and it's not a cube. It is equivalent to 10^(3*n+1)+1, but I don't know if that implies anything terribly interesting.)
That would be a GFN prime for base=10, so it is very unlikely that there is a prime. (probably we know only: 10^(2^n)+1 is prime for n=0,1).

 2009-12-08, 17:18 #48 Mini-Geek Account Deleted     "Tim Sorbera" Aug 2006 San Antonio, TX USA 17·251 Posts Riesel Base 716 Conjectured k = 238 Found Primes: Code: 3*716^2-1 4*716^5-1 5*716^14-1 7*716^1-1 8*716^2-1 9*716^3-1 10*716^1-1 13*716^27-1 15*716^1-1 17*716^2-1 18*716^51-1 19*716^3-1 20*716^78-1 22*716^11-1 24*716^1-1 25*716^3-1 28*716^1-1 30*716^3-1 32*716^228-1 33*716^1-1 35*716^2-1 37*716^39-1 39*716^9-1 42*716^1-1 43*716^5-1 44*716^4-1 47*716^8-1 48*716^1-1 49*716^1-1 50*716^2-1 52*716^11-1 54*716^44-1 55*716^7-1 57*716^5-1 58*716^915-1 59*716^22-1 60*716^2-1 62*716^6-1 63*716^5-1 64*716^1-1 65*716^670-1 68*716^6-1 69*716^26-1 70*716^1-1 72*716^1-1 73*716^1-1 74*716^6-1 75*716^1-1 77*716^2-1 80*716^8-1 82*716^1-1 83*716^2-1 84*716^2-1 85*716^1-1 87*716^6-1 88*716^167-1 90*716^1-1 93*716^1-1 94*716^29-1 97*716^265-1 98*716^216-1 99*716^283-1 102*716^3-1 103*716^5-1 104*716^4-1 108*716^2-1 110*716^150-1 112*716^1-1 113*716^4-1 114*716^18-1 115*716^1-1 119*716^2-1 120*716^2-1 124*716^7-1 125*716^100-1 127*716^1-1 128*716^30-1 129*716^1-1 130*716^15-1 132*716^3-1 135*716^10-1 137*716^2-1 138*716^1-1 139*716^1-1 140*716^80-1 142*716^97-1 143*716^20-1 145*716^3-1 147*716^1-1 148*716^1-1 149*716^6-1 150*716^4-1 152*716^96-1 153*716^1-1 154*716^145-1 158*716^2-1 159*716^1-1 160*716^5-1 162*716^7-1 163*716^1-1 164*716^2-1 165*716^3-1 167*716^2-1 168*716^42-1 169*716^11-1 172*716^3-1 173*716^2-1 174*716^2-1 175*716^1-1 178*716^1-1 180*716^1-1 182*716^20-1 184*716^1-1 185*716^4-1 187*716^313-1 189*716^17-1 192*716^2-1 193*716^419-1 195*716^1-1 197*716^52-1 198*716^1-1 202*716^9-1 203*716^16-1 204*716^1-1 205*716^7-1 207*716^26-1 208*716^1-1 212*716^12-1 213*716^5-1 214*716^5-1 215*716^22-1 217*716^1-1 218*716^4-1 219*716^4-1 220*716^1-1 223*716^1-1 224*716^4-1 225*716^5-1 227*716^8-1 228*716^131-1 229*716^25-1 230*716^16-1 233*716^1972-1 234*716^1-1 237*716^1-1 Remaining k's: Tested to n=2500 Code: 2*716^n-1 29*716^n-1 38*716^n-1 95*716^n-1 107*716^n-1 109*716^n-1 117*716^n-1 123*716^n-1 134*716^n-1 179*716^n-1 190*716^n-1 194*716^n-1 200*716^n-1 Trivial Factor Eliminations: Code: 1 6 11 12 14 16 21 23 26 27 31 34 36 40 41 45 46 51 53 56 61 66 67 71 76 78 79 81 86 89 91 92 96 100 101 105 106 111 116 118 121 122 126 131 133 136 141 144 146 151 155 156 157 161 166 170 171 176 177 181 183 186 188 191 196 199 201 206 209 210 211 216 221 222 226 231 232 235 236 Base Released
2009-12-08, 17:46   #49
Mini-Geek
Account Deleted

"Tim Sorbera"
Aug 2006
San Antonio, TX USA

17·251 Posts

Quote:
 Originally Posted by R. Gerbicz That would be a GFN prime for base=10, so it is very unlikely that there is a prime. (probably we know only: 10^(2^n)+1 is prime for n=0,1).
So we'd need a prime 10^(2^x)+1 that can also be expressed as 10^(3*n+1)+1, right? For that to be, 2^x needs to be 1 mod 3, which means x needs to be even. I think it's safe to say Sierp base 1000 won't be proven by finding a prime in any of our lifetimes, if ever.

Last fiddled with by Mini-Geek on 2009-12-08 at 17:56

2009-12-08, 18:01   #50
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,489 Posts

Quote:
 Originally Posted by Mini-Geek So we'd need a prime 10^(2^x)+1 that can also be expressed as 10^(3*n+1)+1, right? For that to be, 2^x needs to be 1 mod 3, which means x needs to be even. I think it's safe to say Sierp base 1000 won't be proven by finding a prime in any of our lifetimes, if ever.
We can also define Sierpinski/Riesel numbers in a way that we ignore all multipliers that give GFN numbers (my covering.exe used this definition). And in this case the conjecture is proven for this base.

2009-12-08, 18:05   #51
Mini-Geek
Account Deleted

"Tim Sorbera"
Aug 2006
San Antonio, TX USA

17×251 Posts

Quote:
 Originally Posted by R. Gerbicz We can also define Sierpinski/Riesel numbers in a way that we ignore all multipliers that give GFN numbers (my covering.exe used this definition). And in this case the conjecture is proven for this base.
Oh okay, that makes sense. I wonder why the script has a section for k's eliminated because they're equivalent to GFNs, but didn't detect k=10 as such. Perhaps it only looks for GFNs in base b, so it didn't notice that k=10 made a base 10 GFN?

Last fiddled with by gd_barnes on 2010-01-18 at 14:34 Reason: remove base <= 500

2009-12-08, 19:50   #52
gd_barnes

May 2007
Kansas; USA

3×3,499 Posts

Quote:
 Originally Posted by Mini-Geek Oh okay, that makes sense. I wonder why the script has a section for k's eliminated because they're equivalent to GFNs, but didn't detect k=10 as such. Perhaps it only looks for GFNs in base b, so it didn't notice that k=10 made a base 10 GFN?
Now, isn't that interesting? Last night, when testing bases 512 and 1024, I discovered the same bug and believe-it-or-not, it is pretty rare. It only happens on even Sierp bases that are perfect powers where the k's that are the root of the base are not trivial.

That is there are more GFNs than just b^m*b^n+1. There's also q^m*b^n+1 where q is a perfect root of b (base). That is since 1000=10^3, then q=10, so k=10^0, 10^1, 10^2, etc. are also GFNs for base 1000.

I knew this from my experience with base 32, which has GFNs for k's that are powers of 2 (instead of only 32) and completely forgot about it when I made the final modifications to the script.

This shouldn't be hard to change the script. I need to add Willem as one of the main contributors in the comments anyway as well as put some sort of version in there. I'll call Karsten/Micha's original version 1.0, Willem's version 2.0, and Ian/my version 3.0. I'll then make the version with the correct for the GFNs version 3.1.

Gary

Last fiddled with by gd_barnes on 2010-01-18 at 14:35 Reason: remove base <= 500

2009-12-08, 20:39   #53
Mini-Geek
Account Deleted

"Tim Sorbera"
Aug 2006
San Antonio, TX USA

17·251 Posts

Quote:
 Originally Posted by gd_barnes Now, isn't that interesting? Last night, when testing bases 512 and 1024, I discovered the same bug and believe-it-or-not, it is pretty rare. It only happens on even Sierp bases that are perfect powers where the k's that are the root of the base are not trivial. That is there are more GFNs than just b^m*b^n+1. There's also q^m*b^n+1 where q is a perfect root of b (base). That is since 1000=10^3, then q=10, so k=10^0, 10^1, 10^2, etc. are also GFNs for base 1000.
Hm...yeah, I see what you mean, and it does look pretty rare.
Quote:
 Originally Posted by gd_barnes This shouldn't be hard to change the script.
Ok good.

2009-12-08, 22:51   #54
gd_barnes

May 2007
Kansas; USA

244018 Posts

Quote:
 Originally Posted by Mini-Geek Hm...yeah, I see what you mean, and it does look pretty rare. Ok good.
Just to clarify one more thing: Since k=1, 10, 100, etc. are GFN's for Sierp base 1000, as indicated by R. Gerbicz (Robert?), the base is proven.

I noticed that one thing that makes them more rare than expected is that many of the k's that make "non-standard" GFNs for bases are first eliminated by trivial factors and so would not get checked by the GFN routine. It is not possible for "standard" GFNs to be eliminated by trivial factors because standard GFN's can only have the factors of b and trivial k factors are based off of the factors of b-1. By mathematical rule, consecutive numbers cannot have any common factors. So they become an issue immediately and clearly if you don't eliminate them ahead of time and they don't have a prime at a fairly low n-value whereas the non-standard ones take a while to pop their heads up in somewhat unusual situations.

Edit: One more thing I just realized. I need to tweak my definition of GFN's on the web pages and the project definition in the "come join us" thread. Just another thing to do. lol

Gary

Last fiddled with by gd_barnes on 2009-12-08 at 22:59

 2009-12-12, 10:28 #55 gd_barnes     May 2007 Kansas; USA 244018 Posts Reserving Riesel and Sierp bases 512 and 1024. (4 bases) I've already run the script against all 4 to n=2500 but it has the GFN bug so I'm going to use them for testing when I get to that. Outstide of the erroneous GFNs remaining, there's not much remaining on most of them. I've also wanted to kick start a few more power-of-2 bases.

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