Let be a prime dividing . Then
That is: so 17 is a quadratic residue module p. So it follows that
By the Quadratic Reciprocity Theorem: hence it follows that , that is, all primes dividing are quadratic residues module 17, hence so is .
But clearly, reading the equation module 17 we have: since for some integer , coprime to , it follows that: which is impossible since is not the 4th power of an integer module 17. Thus we conclude that the equation can have no solutions.
- an easy way to check this it to remember that ( being coprime to p) has solutions if and only if , in our case n=4 and p = 17 and we see that -
The prime p cannot be 17, because cannot be divisible by 17, otherwise the equation would imply that 17 also divides . Let and then and note that the LHS is dividible by 17² while the RHS is not.