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2010-02-21, 07:16   #12
10metreh

Nov 2008

2×33×43 Posts

Congratulations!

Quote:
 Originally Posted by CRGreathouse My apologies for being 'out of the loop'. But what is known about these so-called Vrba-Reix prps? All Wagstaff primes are VRprps, but is it known that composite VRprps are asymptotically rare?
I think we need to coin a new term: vrobable prime

2010-02-21, 20:24   #13
T.Rex

Feb 2004
France

38F16 Posts

Quote:
 Originally Posted by CRGreathouse Congrats, Tony!
Thanks !
Quote:
 My apologies for being 'out of the loop'. But what is known about these so-called Vrba-Reix prps ?
Look at these explainations.
Quote:
 All Wagstaff primes are VRprps, but is it known that composite VR prps are asymptotically rare?
We have no idea. However, based on my attempt to prove my conjecture for Mersennes, I think that some more conditions are required for a Wagstaff PRP to be a Prime: I think that the product of all S(n) (in: S(n)=S(n-1)^2-2) must be 1 (modulo Wq).
However, I guess that the probability for a Wagstaff composite to have the VR property is VERY VERY small, if non null. Since VR is based on a Cycle that is of length q (Wq=(2^q+1)/3), I think there is no way that a composite can have VR true. But, without a proof... it is only guesses. Any idea for building a proof of VR ?

Read the papers I've gathered on my Blog. It explains many things. If there is something you do not understand, just ask me.

Regards,

Tony

 2010-02-21, 20:39 #14 ixfd64 Bemusing Prompter     "Danny" Dec 2002 California 22·3·191 Posts This is very impressive. I hope it gets proven soon!
 2010-02-23, 08:22 #15 ATH Einyen     Dec 2003 Denmark 1011010011002 Posts Congratulations on the new PRP! Prime95 PRP-test concur :) UID: athath/laptop, 2^4031399+1/3 is a probable prime! Wd4: 2044F973,00000000
 2010-02-23, 08:58 #16 Cybertronic     Jan 2007 DEUTSCHLAND ! 26710 Posts Congratulation for this nice number !!!! 10 years ago, I tried the Lucas Lehmer Test for this type of numbers. Is it a prime, so the result S=14 or S=194 ( S=2 is for Mp's ), but it is maybe not a proof.
2010-02-23, 13:58   #17
T.Rex

Feb 2004
France

911 Posts

Quote:
 Originally Posted by Cybertronic Congratulation for this nice number !!!!
Thanks !
Quote:
 10 years ago, I tried the Lucas Lehmer Test for this type of numbers. Is it a prime, so the result S=14 or S=194 ( S=2 is for Mp's ), but it is maybe not a proof.
Hummm LLT for Mersenne makes use of the Tree of the DiGraph under x^2-2 . With Wagstaff numbers, there is no tree, only cycles. So LLT (meaning: ending with a 0, and then -2 and then 2) cannot work for Wagstaff.
Which seed did you use ? since you only talk of the last step.

Tony

2010-02-23, 13:59   #18
T.Rex

Feb 2004
France

911 Posts

Quote:
 Originally Posted by ATH Congratulations on the new PRP!
Thanks !
Quote:
 Prime95 PRP-test concur :) UID: athath/laptop, 2^4031399+1/3 is a probable prime! Wd4: 2044F973,00000000
Thanks for double-checking !!!!

Tony

2010-02-23, 14:22   #19
Cybertronic

Jan 2007
DEUTSCHLAND !

3×89 Posts

Quote:
 Originally Posted by T.Rex Thanks ! Hummm LLT for Mersenne makes use of the Tree of the DiGraph under x^2-2 . With Wagstaff numbers, there is no tree, only cycles. So LLT (meaning: ending with a 0, and then -2 and then 2) cannot work for Wagstaff. Which seed did you use ? since you only talk of the last step. Tony
n=2^p+1 , not (2^p+1)/3
You start with S=14, after the LL procedure (S=S^2 mod n)-2 you get
S=14 or S=194 (14^2-2).

Here the UBASIC program:
10 P=3
20 N=2^P+1
30 T=1:S=14
40 S=(S^2 mod N)-2
50 inc T
60 if T=P then print P;S:P=nxtprm(P):goto 20
65 if P>100 then stop
70 goto 40

Here the plot:
3 5 -> 5=14-9
5 14
7 65 -> 65=194-129
11 14
13 194
17 14
19 194
23 14
29 76653758
31 194
37 42511304261
41 359734364441
43 194
47 17518790332040
53 6406117628425502
59 508334340273339569
61 194
67 21819313749230765027
71 1195598766420873147536
73 5630004848125169115977
79 194
83 5637954672539490707970155
89 20493566371156403602606220
97 68242359663609183220101332858
101 14
103 435037204291375683415946537180
107 32093918361200282030236449354443
109 414278313959563741717336012443446
113 5112917025335077160694569180088809
127 194
131 1973968232982423701206473030341408459198
137 47484206655905551903295319361957840172753
139 168667007687552910631152078108586066160726
149 18453965156596083542114781548941146664272227
151 262976984875073064821485878363989765425623023
157 49547112554957562135729742000244759963364760238
163 898773787877688180281701171238174776273408405979
167 14
173 6874700042006001666774484698531768324621714679913621
179 297595739783471837689716178237652964409196604508541930
181 1244797157598192014982163301734406121591070982512878662
191 14
193 246557495726160777169800308963987254019574345500178382517
197 154344194312622000819873725710721046832719797408900851567781
199 194

next p's are
313 194
347 14
701 14

The same like #7 List of Wagstaff Primes:
(2) 3 5 7 11 13 17 19 23 31 43 61 79 101 127 167 191 199 313 347 701

best

Last fiddled with by Cybertronic on 2010-02-23 at 14:31

 2010-02-23, 19:26 #20 Cybertronic     Jan 2007 DEUTSCHLAND ! 3·89 Posts short addition : I get with my program: OK list 10 P=3 20 N=2^P+1 30 T=1:S=14 40 S=(S^2 mod N)-2 50 inc T 60 if T=P then goto 100 70 goto 40 100 if or{S=(14 mod N),S=(194 mod N)} then print P; 120 P=nxtprm(P):goto 20 OK run 3 5 7 11 13 17 19 23 31 43 61 79 101 127 167 191 199 313 347 701 1709 2617 3539 5807 exact all wagstaff prime exponents in logical order... Last fiddled with by Cybertronic on 2010-02-23 at 19:30
 2010-02-24, 15:46 #21 Jeff Gilchrist     Jun 2003 Ottawa, Canada 7·167 Posts For fun I ran this exponent through my own GMP based Vrba-Reix PRP code on an Opteron and Core2 to verify and it comes out as prime: W4031399 is prime! W4031399 = ...2388889642104198122875563 (1213572 digits) Finished! Checked prime q values to: 4031399 Great work DUR! Jeff.
2010-02-24, 21:35   #22
T.Rex

Feb 2004
France

911 Posts

Quote:
 Originally Posted by Jeff Gilchrist .... and it comes out as prime:
As a PRP. Sure it is a prime... but we lack a proof. So it is a PRP.
Thanks for the nice job, Jeff !!
Tony

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