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 20120331 10:29:30 