Hi folks
>Mp is a Mersenne Prime with odd prime p "iff"
>3^((Mp-1)/2)=-1 (mod Mp) .
>**********
>At first sight, I thought "that's not right", but a few minutes of testing
a
>few Mersenne primes and non-prime Mersenne numbers on my TI-92+ has upheld
the
>"iff". Can anyone find a counterexample? This is bugging me. Augh!
>S.T.L.
Well one way is easy. The calculation is actually the quadratic residue
symbol (Euler pseudoprime test) for 3. Since (q|3) depends on q mod 12, and
all Mersennes of odd exponent>=3 are =7 mod 12, then if Mp is prime, the
statement is true.
Conversely if the statement above is true, then Mp is a strong probable
prime to base 3. Now composite probable primes do exist (though they get a
bit thin on the ground), more importantly there are an infinity of
counterexamples given any number of strong probable primality tests. So the
result is not necessarily true *but* a counterexample may be very difficult
to find. However, it's much better than strong probable primality to base 2,
*all* Mersennes of prime exponent, Mp prime or not, are strong probable
prime to base 2.
Most importantly though, the computation required to perform this
calculation is no less than a Lucas-Lehmer test, which gives an absolute yes
or no result.
Chris Nash
Lexington KY
UNITED STATES
=======================================================
Co-discoverer of the 7th and 10th largest known primes.
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