Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
Mersenne primes
Replies:
6
Last Post:
May 23, 2001 3:26 PM




Re: Mersenne primes  meet Wieferich primes
Posted:
May 21, 2001 8:52 AM


Zundark wrote: > > carel wrote: > > > I recently saw that 2^71 1 has not been tested for primeness. > > I can't imagine where you saw this. 2^p  1 has been tested > at least once for all p < 6434700. See the GIMPS status page: > http://www.mersenne.org/status.htm . If you really want > to search for new Mersenne primes you should join GIMPS. >  Zundark
A propos de Mersenne primes: M(2,p) = 2^p 1 (prime exponent p) has binary representation: p ones.
Generalizing this to higherbase representations yields: p ones (base b) M(b,p) = (b^p 1)/(b1).
Notice the only known two Wieferich primes 1093 and 3511 are:
base b=3, p=7: 1093 = 1111111 = (3^7 1)/2 = M(3,7)
base b=8, p=4: 3511 = 6667 = (8^4 1)(6/7) +1 = 6M(8,4) +1
Any suggestion why these are the only known primes p < 4.10^12 with 2^p = 2 (mod p^2)  Is this necessary or coincidence ?)
 NB  http://arXiv.org/abs/math/0103067 "On primitive roots of 1 mod p^k, divisors of p^2 1, and FST* mod p^3"
 Math = The Art of separating Necessity from Coincidence 



