If you can work with modular arithmetic, then you're done.
You have to prove that there is no k such that
Fermat's little theorem tells us that
so consider the possible values of k modulo 6 :
As you say, we will try to find a contradiction. First note that
The fact that implies since
Now implies thus and then ( In general we have that if and only if )
We then write that is which is not possible since and so (expand using the binomial theorem)