Primorials

Date: 10/15/2003 at 14:45:49
From: Ryan 
Subject: the proof of infinite primes

So we know that p_1 * p_2 * ... * p_n + 1 is either prime or divisible
by a prime not included in the list. 

I can't find an example where the result is NOT prime, which makes me
wonder whether the second part ("is divisible by a prime not included
in the list") is superfluous.

Do you know an example of one?

Date: 10/15/2003 at 16:58:11
From: Doctor Peterson
Subject: Re: the proof of infinite primes

Hi, Ryan.

We should first note that it isn't necessary that this ever actually 
happens; it's just part of the logic of the proof that we can't show 
that p_1 * p_2 * ... * p_n + 1 must be prime, but we _can_ show that
if it isn't itself a new prime, it must be divisible by a new prime.
The proof is perfectly valid even if we can never find any case where
the latter is true.

But it's not hard to find one, if you have the patience or technology 
to factor big numbers. Here are my results:

  2+1 = 3 prime
  2*3+1 = 7 prime
  2*3*5+1 = 31 prime
  2*3*5*7+1 211 prime
  2*3*5*7*11+1 = 2311 prime
  2*3*5*7*11*13+1 = 30031 = 59*509

You can see a list of which numbers of this form ARE prime here:

  Primorial
  http://mathworld.wolfram.com/Primorial.html 

    p# + 1 is known to be prime for the primes p = 2, 3, 5, 7, 11, 31,
    379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, ... 

where p# means the product of all primes up to and including p.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/