Sum of Two Different Primes

Date: 02/22/2002 at 10:36:26
From: Nate
Subject: Prime numbers

Can the sum of two different primes ever be a factor of the product 
of those two primes? I.e., Do there exist primes p and q such that 
(p+q)/pq?

Date: 02/22/2002 at 23:00:10
From: Doctor Paul
Subject: Re: Prime Numbers

There do not exist distinct primes p and q such that (p+q) divides the 
product p*q.  To see why this must be the case, think about the 
divisors of p*q. They are 1, p, q, and p*q. Clearly p+q is greater 
than 1, p, and q, so the only way that p+q could be a factor of p*q 
would be if p+q = p*q.

Divide both sides of this equation by p*q to obtain:

   1/q + 1/p = 1

If p is the smaller of the two primes, then the smallest p can be is 
two, and the smallest q can be is three. In this case,

   1/q + 1/p = 1/3 + 1/2 = 5/6 < 1

Any other choices of p and q will necessarily be larger, which will 
make the sum 1/q + 1/p even farther away from one. Thus 1/q + 1/p can 
never be one if p and q are distinct primes.

This completes the proof.

I hope this helps.  Please write back if you'd like to talk about this 
more.

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