2003 IMO Problems/Problem 6

Problem

p is a prime number. Prove that for every p there exists a q for every positive integer n, so that $n^p-p$ can't be divided by q.

Solution

Let N be $1 + p + p^2 + ... + p^{p-1}$ which equals $\frac{p^p-1}{p-1}$ $N\equiv{p+1}\pmod{p^2}$ Which means there exists q which is a prime factor of n that doesn't satisfy $q\equiv{1}\pmod{p^2}$ . \\unfinished

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2003 IMO Problems/Problem 6

2003 IMO Problems/Problem 6

Problem

Solution