> Someone asked if there could be a better method for testing
> Mersenne numbers
> than the L-L test. I think that the L-L test is essentially
> as efficient as
> possible. However, I think that there is room for improvement in the
> algorithms for doing the arithmetic involved.
That is a remarkably bold claim. It's akin to saying that the only way of
finding primes is by brute force testing of all candidates.
To me, at least, it is conceivable that some advance in number theory will
prove that there is a finite number of Mersenne primes and that their values
are given by a relatively simple formula. Consider the history of FLT
(number of solutions of a^n+b^n=c^n) for an analogous case. Initially, each
exponent was tested in turn; then it was shown that an infinite class of
exponents could not be a solution; finally it was shown that only a finite
number of exponents could be a solution and, in particular, only the
exponents 0, 1 and 2 were valid. The final result used non-trivial
mathematics but did not require large amounts of numerical computation.
Paul
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