Writing 31 as sum of 15 fourth powers

Hi everyone,

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?

Thanks.

Re: Writing 31 as sum of 15 fourth powers

all polynomials are continuous but that is only when smaller rational and irrational numbers are included, if they are discounted above then it will have infinite discontinuities. All polynomials are discontinuous if rationals and irrationals are excluded

Re: Writing 31 as sum of 15 fourth powers

I might start by considering integers "N" that can be written as 2y^4 - 1 for any positive integer y > 1.

Ex. Notice 31 = 2 x (2^4) - 1