Re: Space between prime numbers

From: Randy Poe (poespam-trap_at_yahoo.com)
Date: 03/28/05 

Date: 28 Mar 2005 07:15:21 -0800

Richard Cavell wrote:
> Hi,
>
> Given any prime number, I wonder if there is a way to predict what
the
> gap will be between it and the next prime number.

Not that I've ever heard of. If there were, it would be easy
to set new "largest prime" records.

>
> Obviously you can make good guesses (only examine odd numbers, etc)
and
> then you can test for primality.

You can make guesses. Beyond oddness, I'm not sure what
property you can choose in advance to make it more likely
to be prime, except perhaps for choosing Mersenne numbers,
numbers of the form 2^p - 1 where p is prime. It is known
that infinitely many Mersenne numbers are prime, but
infinitely many Mersenne numbers are composite, and there
are lots of non-Mersenne primes also of course.

> It is said that prime numbers are irregularly spaced.

First define "irregularly spaced" a little more precisely.

There are various theorems about spacing. I think it
is proven that there exist arbitrarily long sequences
of sequential odd numbers (e.g. 3,5,7) among the primes,
and I think it's also proven that all (even) spacings no matter
how large occur somewhere among the primes (e.g., there
are sequential primes somewhere which are separated
by 12345678910).

> How does one
> prove this? Obviously the sieve of Eratosthenes implies there is
indeed
> a pattern to the prime numbers,

It does? It's not obvious to me. How does it imply there's
a pattern?

             - Randy

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