prove that 1600010 cannot be written as a sum of nine perfec

prove that 1600010 cannot be written as a sum of nine perfect fourth powers. (hit: consider this situation mod 16)

Solution

First, take some arbitrary integer x.

If x is even, then x = 2k for some integer k.
Then x^4 = (2k)^4 = 16k, which is divisible by 16.

If x is odd, then x = (2k+1), for some integer k.
(2k+1)^4 = 16k^4 + 32k^3 + 24k^2 + 8k + 1
Clearly, 16k^4 and 32k^3 are divisible by 16.
Now look at the next two terms: 24k^2 + 8k = 8k(3k+1).
If k is even then 8k is a multiple of 16, so 8k(3k+1) is divisible by 16.
If k is odd then 8k is divisible by 8, and (3k+1) is even, so 8k(3k+1) is divisible by 16.
Hence, if x is odd, then x^4 mod 16 = 1.

From the above, we see that the fourth power of any integer, mod 16, is always 0 or 1.

1600010 mod 16 = 10.
You can\'t add nine numbers that are all zeros or ones, and get a sum of ten.
Hence, 1600010 cannot be written as a sum of nine perfect fourth powers.