mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2006-02-21, 20:12   #12
Citrix
 
Citrix's Avatar
 
Jun 2003

32·52·7 Posts
Default

Cant you use PFGW and find all the primes of this form?

Citrix
Citrix is offline   Reply With Quote
Old 2006-02-21, 20:22   #13
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

11101001001002 Posts
Default

Quote:
Originally Posted by akruppa
This is, in fact, a good question and well posed. Promoted to "Math". Unfortunately, I don't know the answer.

Alex
Almost certainly there are finitely many.

We want N= 2^p-1 and N1 = 2^(p-1)(2^p - 1) + 1 to be simultaneously
prime. The probability of this is O(1/ ((log N)(log N1))) and the sum over
all p converges. It converges fairly rapidly.
R.D. Silverman is offline   Reply With Quote
Old 2006-02-21, 21:47   #14
akruppa
 
akruppa's Avatar
 
"Nancy"
Aug 2002
Alexandria

2,467 Posts
Default

Agreed. For all known Mersenne primes we now know that there aren't any more primes of this form and larger such primes are extremely unlikely. So the answer to the original poster's question is: almost certainly not.

As for the question for an efficient primality proof: since we have the complete factorisation of P(n) by definition of P(n) (for n a Mersenne prime exponent), a simple P-1 primality test will work. Find some integer a so that a[I]P[/I]([I]n[/I])≡1 (mod P(n)+1) and for each p|P(n), a[I]P[/I]([I]n[/I])/[I]p[/I]!≡1 (mod P(n)+1). The test can be further simpified but this is the basic idea.

Alex
akruppa is offline   Reply With Quote
Old 2006-02-25, 20:00   #15
flava
 
flava's Avatar
 
Feb 2003

2×59 Posts
Default

I looked for the primes satisfying your equation and for the following primes p (2^p-1)*(2^(p-1))+1 is prime or probable prime:

p in {2,3,13,19,271,601,4729,34039}

Tested up to p=67261

As you can see they are quite scarce ...

Last fiddled with by flava on 2006-02-25 at 20:01
flava is offline   Reply With Quote
Old 2006-02-27, 12:01   #16
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22×5×373 Posts
Default

Quote:
Originally Posted by flava
I looked for the primes satisfying your equation and for the following primes p (2^p-1)*(2^(p-1))+1 is prime or probable prime:

p in {2,3,13,19,271,601,4729,34039}

Tested up to p=67261

As you can see they are quite scarce ...
This is not the original sequence. For the sequence you propose, we
expect infinitely many. The original sequence ALSO requires that 2^p-1
is prime.
R.D. Silverman is offline   Reply With Quote
Old 2006-02-27, 21:23   #17
flava
 
flava's Avatar
 
Feb 2003

2×59 Posts
Default

Quote:
Originally Posted by R.D. Silverman
This is not the original sequence. For the sequence you propose, we expect infinitely many. The original sequence ALSO requires that 2^p-1 is prime.
Yes I know. I was just satisfying my own curiosity and made this quick check to see if the distribution of (2^p-1)*(2^(p-1))+1 primes is not higher than expected. What makes you expect infinitely many primes of this form? Is it the fact that sum(1/p) diverges over primes sufficient?
flava is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Distribution of Mersenne primes before and after couples of primes found emily Math 34 2017-07-16 18:44
Primes in n-fibonacci sequence and n-step fibonacci sequence sweety439 And now for something completely different 17 2017-06-13 03:49
Primes in A048788 OEIS Sequence carpetpool Miscellaneous Math 9 2017-03-17 22:57
Sequence of primes formation PawnProver44 Homework Help 37 2016-03-19 15:43
Will GIMPS Ever Discover Two Primes Out of Sequence? jinydu Lounge 11 2009-06-06 16:40

All times are UTC. The time now is 15:28.

Wed Dec 2 15:28:05 UTC 2020 up 83 days, 12:39, 2 users, load averages: 2.67, 3.19, 2.77

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.