More on sums of squares?

In the thread about distinct distances I realised that the problem was
really about sums of squares and the complete solution to that was
provided. I looked briefly for more information online and couldn't
find it. I would like to comment further.

It is said that almost all numbers are not the sum of two squares.
This is rather surprising. On probablistic arguments, one concludes
that the fraction should be non-zero, and also sums of k kth powers
for any k. So is this true for other k, that almost all numbers are
not the sum of k kth powers?

I can follow the logic, though, of why this is true for squares. The
part that I've never been able to get is why all 4k+1 primes must be
so expressible - it seems perfectly conceivable to me that there's a
prime that is not such a sum.

Andrew Usher

.

Relevant Pages
Quantcast