Bases 501-1030 reservations/statuses/primes - Page 5

TimSorbet · 2009-12-08, 14:43

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

TimSorbet · 2009-12-08, 14:50

Riesel Base 713
Conjectured k = 8

Found Primes:

Code:

2*713^2-1
4*713^1-1
6*713^2-1

Conjecture Proven

R. Gerbicz · 2009-12-08, 16:05

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).

TimSorbet · 2009-12-08, 17:18

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:

Base Released

TimSorbet · 2009-12-08, 17:46

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.

R. Gerbicz · 2009-12-08, 18:01

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.

TimSorbet · 2009-12-08, 18:05

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?

gd_barnes · 2009-12-08, 19:50

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

TimSorbet · 2009-12-08, 20:39

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.

gd_barnes · 2009-12-08, 22:51

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

gd_barnes · 2009-12-12, 10:28

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.

2009-12-08, 14:43	#45
TimSorbet Account Deleted "Tim Sorbera" Aug 2006 San Antonio, TX USA 1000010110111₂ Posts	Riesel Base 707 Conjectured k = 14 Found Primes: Code: 2707^350-1 4707^3-1 6707^1-1 8707^4-1 10707^1-1 Remaining k's: Tested to n=2500 12707^n-1 Base Released

2009-12-08, 14:50	#46
TimSorbet Account Deleted "Tim Sorbera" Aug 2006 San Antonio, TX USA 10267₈ Posts	Riesel Base 713 Riesel Base 713 Conjectured k = 8 Found Primes: Code: 2713^2-1 4713^1-1 6*713^2-1 Conjecture Proven

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2009-12-12, 10:28	#55
gd_barnes "Gary" May 2007 Overland Park, KS 10111100001010₂ 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.