...
> >The number n=4^7057-3 has been proved prime by cyclotomy: with 4249
> >decimal digits, it is currently the largest prime proved with a
> >general prime proving algorithm. The main stage of the proof took 6
> >hours, the final "Lenstra - gcd and trial division" step (allowing a
> >factored part of O(n^{1/3}) took roughly 2 days.
>
> Luke invited me to try the Meganet program on 4^7057-3. It reported
> the number as prime in 33 minutes on my PPro 200, with a bunch of
> other apps going at the same time.
A strong pseudo prime test is wrong at most 1/4 of the time. However,
in practice this is much smaller. If I use 100 random bases and it
reports a number is prime then the probability it is wrong is
(1/4)^100=6.2E-61. You would need to test more than 6.2E61 values before
having a good probability of finding a counterexample. If random bases
are used even then your counterexample would not be reproducible (without
knowing the random bases). See Knuth v.2 3rd ed. p. 395.
I could disguise the algorithm with some complex arithmetic operations
and call it a "U-sequence". Without publishing the algorithm for
peer-review and a rigorous proof I could claim it was a fast 100%
determinstic primality test. Noone would be able to find a
counterexample.
Perhaps Meganet has done this or they have unknowingly created a
probabilistic primality test.
In Dr. Milstein's endorsement he states "I did not develop rigorous
proofs...I applied the assertions of the paper to a number of non-trivial
values." Regardless of his credibility this endorsement is meaningless.
It is just marketing hype for their primality test.
Without a proof Meganet's primality test is still a probabilistic test,
so why not use a free one. Here is one, no marketing hype, no
endorsements and no "T-sequences"! Compile with GNU-MP.
#include <stdio.h>
#include "gmp.h"
int main() {
mpz_t N;
mpz_init (N);
printf ("Enter number:\t");
mpz_inp_str (N, stdin, 10);
if (mpz_probab_prime_p (N, 100)) printf ("Prime\n");
else printf ("Composite\n");
return (0);
}
>
> I had planned to get the code before asking the list for a few folks
> interested in taking a crack at finding a flaw in it. We get only
> three evaluations under NDA. Maybe we can use one up to hook it up on
> a server under a web form in kind of constrained batch mode. Any
> takers? Please email me privately.
Ask to see their proof and not just the code. Their website claims they
have one.
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