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-   -   Bases 501-1030 reservations/statuses/primes (https://www.mersenneforum.org/showthread.php?t=12994)

 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
[/code]Remaining k's: Tested to n=2500
12*707^n-1

Base Released

 TimSorbet 2009-12-08 14:50

Riesel Base 713

Riesel Base 713
Conjectured k = 8

Found Primes:[code]2*713^2-1
4*713^1-1
6*713^2-1[/code]

Conjecture Proven

 R. Gerbicz 2009-12-08 16:05

[QUOTE=Mini-Geek;198183]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[/CODE]

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.)[/QUOTE]

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
[/code]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[/code]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[/code]Base Released

 TimSorbet 2009-12-08 17:46

[quote=R. Gerbicz;198208]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).[/quote]
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=Mini-Geek;198227]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.[/QUOTE]

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=R. Gerbicz;198234]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.[/quote]
Oh okay, that makes sense. :smile: 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=Mini-Geek;198236]Oh okay, that makes sense. :smile: 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?[/quote]

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=gd_barnes;198250]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.[/quote]
Hm...yeah, I see what you mean, and it does look pretty rare.
[quote=gd_barnes;198250]This shouldn't be hard to change the script.[/quote]
Ok good. :smile:

 gd_barnes 2009-12-08 22:51

[quote=Mini-Geek;198259]Hm...yeah, I see what you mean, and it does look pretty rare.

Ok good. :smile:[/quote]

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. :smile:

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