Date: Dec 28, 2012 10:14 PM
Author: kirby urner
Subject: Re: An Interesting Way To Introduce Prime Numbers
On Fri, Dec 28, 2012 at 10:35 AM, Ken Abbott <abbottsystems@gmail.com> wrote:

> http://www.math-math.com/2012/12/making-prime-numbers.html

Yes, well described.

In practice it's not always easy to determine which prime factors make

N, so one successively tries those on file.

Like when you asked about 15, 2 doesn't work so we know to try 3, but

sometimes it's not obvious with which odd prime to start, so one plows

through the ones on file in succession, up to sqrt(N), at which point

one should have found one if there is one. If not, add to the stash

and proceed.

Put another way, factoring is a non-trivial problem and once the

integers get big enough, we run out of memory and/or time given

present capacity.

Which is not to denigrate your algorithm. The algorithms giving us

primes of many thousands of digits tend to shoot for "probable primes"

which means very very unlikely to be composite. Composites sneak

through the filters.

Like remember the Carmichael numbers: they pass all the Fermat tests

suggesting they're prime, but they're not:

http://crypto.stanford.edu/pbc/notes/numbertheory/carmichael.xhtml

http://crypto.stanford.edu/pbc/notes/numbertheory/millerrabin.xhtml

Your method has often been implemented and is well respected. It

requires a growing stash of primes on file.

Kirby