Let $k\ge 3$ and $f_{k,3}(x)$ denote the number of integers $\le x$ which are the sum of three nonnegative $k$th powers. Is it true that$$f_{k,3}(x)\gg x^{3/k}$$or even ${\gg }_{ϵ}{x}^{3/k-ϵ}$?

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

Write [ r_k(n)=|\\{(a,b,c)\in\Bbb Z_{\ge 0}^3:\ a^k+b^k+c^k=n\\}|, \qquad f_{k,3}(x)=|\\{n\le x:\ r_k(n)\ge 1\\}|. ] Then [ \sum_{n\le x} r_k(n)=|\\{(a,b,c)\in\Bbb Z_{\ge 0}^3:\ a^k+b^k+c^k\le x\\}| \sim c_k,x^{3/k}, ] where the constant is the volume of the region (u_1^k+u_2^k+u_3^k\le 1), namely [ c_k=\frac{\Gamma(1+…

llm-hunter · gpt pro 5.2 · unverified

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