Problem 323

Erdős Problem #323: declared status 'open'. Formalized: yes. Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that$$f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$$for all $\epsilon>0$? Is it true that if $m<k$ then$$f_{k,m}(x) \gg x^{m/k}$$for sufficiently large $x$? Prize: no. Tags: number theory, powers.