[giik]

I think it's even simpler after you figured out that m = kn = n^k => n is an integer > 1 and equal to the (k-1)-th root of k where k is an integer > 1.

The only integer (k-1)-th root of k is 2 for k = 2. Thus, n = 2, k = 2, m = 4.

[bustermellotron]

The “claim” more or less proves that k-1 th root of k is less than 2 if k is larger than 2. So I think your argument is equivalent.