The solution is
Take modulo we have
otherwise if is odd we have which is impossible .
Thus we have
which leads to
where
which is the multiple of if which is again impossible .
Thus ( not because ) but they can't be both zero as we are looking for positive integers
We now show that is the only solution .
Suppose so we have
If we write down the first six powers of modulo , we obtain :
we have exactly and exactly
We conclude that for some non-negative integers
Back to this equation :
write we have
the multiple of which is also impossible .
Therefore ,
is the only hope , luckily , we obtain and , is the only solution .