The proof for the case n = 4 also proves the cases n = 8, 12, 16, 20,….The reason is that any number that can be written as an 8th (or a 12th, 16th, 20th,…) power can also be rewritten as a 4th power. For instance, the number 256 is equal to 28, but it is also equal to 44. Therefore any proof that works for the 4th power will also work for the 8th power and for any other power that is a multiple of 4. Using the same principle, Euler’s proof for the case n = 3 automatically proves the cases n = 6, 9, 12, 15,…
