Let $1\le m\le k$ and $f_{k,m}(x)$ denote the number of integers $\le x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that$$f_{k,k}(x){\gg }_{ϵ}{x}^{1-ϵ}$$for all $ϵ>0$? Is it true that if $m<k$ then$$f_{k,m}(x)\gg x^{m/k}$$for sufficiently large $x$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write [ f_{k,m}(x)=|\\{n\le x:\ n=a_1^k+\cdots+a_m^k\ \text{for some }a_i\in\mathbb Z_{\ge 0}\\}|. ]

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.