Prove that for each prime p, there exists a prime q such that n^p − p is not divisible by q for any positive integer n.

Source: IMO 2003

I think this is the shortest way to do it.

Say p=2, then q=3 qualifies as required.         
         
So p and q are odd.         
         
Let P denote p(x), the xth odd prime, then n^P−P is equivalent to the expression given         
         
Let q be the smallest prime of the form (2k*P+1); there must be one.         
         
Now by substitution we have (n^P − P)/(2k*P+1), with no integral solutions: e.g. small values of {P, q} {2,3}{3,7}{5,11}{7,29}{11,23} etc, compare my own problem, Sweet 17

Edited on March 31, 2012, 10:55 am

Posted by broll on 2012-03-31 10:29:30