> Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the su

> Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low. You're mixing up "N is an nth power" with "there exists an N that is an nth power". The purpose over summing over all the integers is to obtain the latter from the former, and it's not necessarily very small even if the former is small.