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Topic: Mersenne primes
Replies: 6   Last Post: May 23, 2001 3:26 PM

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Nico Benschop

Posts: 1,708
Registered: 12/6/04
Re: Mersenne primes - meet Wieferich primes
Posted: May 21, 2001 8:52 AM
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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 higher-base representations yields: p ones (base b)
M(b,p) = (b^p -1)/(b-1).

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 ------







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