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From: dik@cwi.nl (Dik T. Winter)
Subject: Re: 9, prime gone bad.  was RE: zero blah blah
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References: <DMGvo2.Bpo@midway.uchicago.edu> <4fdpif$ovr@muir.math.niu.edu> <4fdvcs$s9e@status.gen.nz>
Date: Fri, 9 Feb 1996 00:03:00 GMT
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Xref: glinda.oz.cs.cmu.edu sci.physics:170251 comp.ai:36870 comp.ai.philosophy:37569 sci.philosophy.meta:24477 sci.math:135640

In article <4fdvcs$s9e@status.gen.nz> ericf@central.co.nz (Eric Flesch) writes:
 > rusin@washington.math.niu.edu (Dave Rusin) wrote:
 > >This is false. There are methods of determining the primality of an
 > >integer which are much faster and which, in particular, do not imply
 > >any knowledge of the factors if the number is indeed shown to be
 > >composite. Just to give a simple and useful example, if  2^n (mod n)
 > >isn't  2, then  n  isn't prime.
 > 
 > All you have demonstrated is that the determination can be done
 > quickly for SOME numbers, i.e. those which can be quickly falsified.
 > You have not demonstrated that prime numbers can be quickly verified
 > as being prime.

But it can.  Much faster than trying to determine the factors.  There
are deterministic methods that will give proof that a number is prime or
not without trying to find any factor, and very much faster.  I have a
program here (*) that will tell you in a few minutes whether a 200-digit
number is prime or not, and it is *not* probabilistic.

As far as I know all factor finding methods are non-deterministic, except
division by succesive primes.  That is, all numbers succumb within the time
estimated for the other methods (which is very long for 200-digit numbers)
by the order-formula's, but as far as I know there is no proof that all will.
But perhaps Bob Silverman or Peter Montgomery know better.

Also it is known that if the Riemann hypothesis holds that primality testing
is even much cheaper than it is now, while it does not help very much in
factor finding.
--
(*)  The program is by Lenstra and Cohen with some of the basics in it done
by me.  A short introduction about it can be found in some back issues of
Mathematics of Computation (around 1986 say).  There have been later
improvements that make primality testing of 1000-digit numbers doable.
Forget factorization of those numbers for the time being.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924098
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
