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 2010-10-26, 11:04 #1 kurtulmehtap   Sep 2009 22×32 Posts mersenne prime as a factor of another number I wonder if there are a special class of number which have mersenne primes as factors? Thanx...
2010-10-26, 11:50   #2
R.D. Silverman

Nov 2003

26·113 Posts

Quote:
 Originally Posted by kurtulmehtap I wonder if there are a special class of number which have mersenne primes as factors? Thanx...
No.

2010-10-26, 12:29   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

8,369 Posts

Quote:
 Originally Posted by R.D. Silverman No.
r.d. silverman are you forgetting perfect numbers ? if so I hope you refresh.

2010-10-26, 13:15   #4
CRGreathouse

Aug 2006

172B16 Posts

Quote:
 Originally Posted by kurtulmehtap I wonder if there are a special class of number which have mersenne primes as factors? Thanx...
Only Mersenne primes, or at least one Mersenne prime?

For the former, that's all subsets of 1 U A056652. For the latter... that's pretty dense, natural density 0.4514311155... if I'm not mistaken. The sequence starts 3,6,7,9,12,14,15,18,21,24,27,28,30,31,... and includes sequences like 22n - 1.

 2010-10-26, 16:41 #5 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 3·419 Posts 23209 is factor of M967 Or do you rather mean that list of Wieferich primes? such as that cases for that of q2 | 2p-1 1093, 3511 For example, please notice that 10932 indeed divides up with that Wagstaff number: (2182+1)/3 Really following up within that way 10932 | M1092 35112 | M3510 of course for ever
2010-10-27, 19:22   #6
ewmayer
2ω=0

Sep 2002
República de California

9,791 Posts

Quote:
 Originally Posted by kurtulmehtap I wonder if there are a special class of number which have mersenne primes as factors? Thanx...
Yes there is:

Initialize x[0] = 4 (other values are also possible, but we'll keep it simple for now)
For m := 2^p-1 prime, do p-2 of the following iterations:
x[i] = x[i-1]^2 - 2

Then x[p-2] is divisible by m.

Simplest case: p=3, m=7, and x[p-2] = x[1] = 14, which is divisible by 7.

2010-10-27, 20:25   #7
R.D. Silverman

Nov 2003

161008 Posts

Quote:
 Originally Posted by ewmayer Yes there is: Initialize x[0] = 4 (other values are also possible, but we'll keep it simple for now) For m := 2^p-1 prime, do p-2 of the following iterations: x[i] = x[i-1]^2 - 2 Then x[p-2] is divisible by m. Simplest case: p=3, m=7, and x[p-2] = x[1] = 14, which is divisible by 7.
One can construct many artificial examples of this type. But I took the
question to mean whether there is some a priori interesting set of numbers divisible by Mersenne primes. One can always construct
such classes. Perfect numbers are an example of such a constructed
class (and are the original reason for the study of M_p).

2010-10-30, 21:26   #8
Damian

May 2005
Argentina

2×3×31 Posts

Quote:
 Originally Posted by ewmayer Yes there is: Initialize x[0] = 4 (other values are also possible, but we'll keep it simple for now) For m := 2^p-1 prime, do p-2 of the following iterations: x[i] = x[i-1]^2 - 2 Then x[p-2] is divisible by m. Simplest case: p=3, m=7, and x[p-2] = x[1] = 14, which is divisible by 7.
I know this is the LL primality test for mersenne numbers (numbers of the form $2^p - 1$ for some prime number $p$ )

Is there a similar primality test for numbers of the form $3^p - 2^p$ for some prime number $p$ ?

Thanks.

Last fiddled with by Damian on 2010-10-30 at 21:26

2010-10-31, 19:49   #9
lavalamp

Oct 2007
London, UK

101000111002 Posts

Quote:
 Originally Posted by Damian Is there a similar primality test for numbers of the form $3^p - 2^p$ for some prime number $p$ ?
I have absolutely no idea, but here are the first 19 values of p where the expression is prime:
Code:
2
3
5
17
29
31
53
59
101
277
647
1061
2381
2833
3613
3853
3929
5297
7417
Checked all p < 30000. Note that the last one is only a probable prime, the others have been deterministically tested with Primo.

 2010-10-31, 20:12 #10 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 23A816 Posts A057468 is well studied.
2010-10-31, 20:55   #11
lavalamp

Oct 2007
London, UK

130810 Posts

Quote:
 Originally Posted by http://www.research.att.com/~njas/sequences/A057468 Some of the larger entries may only correspond to probable primes. The 1137- and 1352-digit values associated with the terms 2381 and 2833 have been certified prime with Primo. - Rick L. Shepherd
I have uploaded primality certificates for p = 3613, 3853, 3929 and 5297, and the certificate for p = 7417 will be uploaded when it's finished. Don't think I'll be creating any for p = 90217 or above...

http://2721.hddkillers.com/3^n-2^n/

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