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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

22·5·373 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

26·131 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

3×1,993 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

22·3·7·139 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

164448 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
Manchester, UK

136210 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 59×163 Posts A057468 is well studied.
2010-10-31, 20:55   #11
lavalamp

Oct 2007
Manchester, UK

101010100102 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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