On 4 Nov 99, at 20:37, Preda Mihailescu wrote:
> Sorry to contradict you, but this things are called ``deterministic under the
> ERH'', which is totally different from deterministic, straight.
Of course, you're right.
> Simply, if the ERH
> is proved one day to be true, all the effort spent meanwhile for finding
> deterministic primality proofs [independent of the ERH] would look
> rather curly. But such is life.
[Phrase in brackets is my insertion - BJB]
Not neccessarily. Work applied in one direction sometimes proves
useful somewhere else, even if the original project fails or is
superceded.
> I was in Rome this summer when the proof of the Taniyama conjecture
> was announced, and at the same conference, in fact the day after the
> announcement, there were several talks about ``elliptic curves which
> are modular'', or ``Weyl curves'' - which were in that day known to be
> simply all elliptic curves :-).
It's unfair to expect people to rewrite their papers overnight!
> And as for your claim that Miller's proof would feature a contradiction to the
> ERH, that is also false.
If we find a number with a certificate of primality derived from some
method not dependent on the ERH but which nevertheless passes
Miller's Test as a strong pseudoprime for all bases less than
2(ln n)^2, then either Miller's paper is broken, or we have a
counterexample disproving the ERH. So far as I'm aware, Miller's
paper is no more contentious than Pythagoras's Theorem.
Proth's Theorem allows us to find certified prime numbers of a fair
size (thousands of digits) quite rapidly. Granted these are a very
thin subset of all numbers, and, even then, running sufficient
Miller's Tests on a number with "only" a thousand digits would take a
very large amount of CPU time. That's why I won't be holding my
breath!
Regards
Brian Beesley
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