I'm supposed to show that you cannot write 31 as a sum of 15 fourth powers (that is, 0^4,1^4, etc).
I have proven this by noting that I can only allow one 2^4 in the expansion at most. I then considered some cases and it turned out to work.
However, I'm now supposed to show that there exist infintely many numbers that cannot be written as a sum of 15 fourth powers. I think I need to prove the first part differently (probably considering 31 = x1^4 +x2^4 + ... x_15 ^4 modulo some integer), but I don't see how.
Can anyone help?