If n epsilon Z then n4 8m or n4 8m 1Solutionn can only be

If n epsilon Z, then n^4 = 8m or n^4 = 8m + 1.

Solution

n can only be odd or even, nothing else.

Let n is even:

So, n = 2k, for some k
n^4 = (2k)^4
= 16k^4
= 8 * (2k^4)
Let 2k^4 = m
So,
n^4 = 8m

If n is odd:
n = 2k + 1, for some k
n^4 = (2k + 1)^4
= 1(2k)^4 + (4 * (2k)^3) + 6 * (2k)^2 + 4 * (2k) + 1
= 16k^4 + 32k^3 + 24k^2 + 8k + 1
= 8(2k^4 + 4k^3 + 3k^2 + k) + 1
Let (2k^4 + 4k^3 + 3k^2 + k) = m
So,
n^4 = 8m + 1