Let $k\ge 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that$$1_A^{(k)}(n)>n^c?$$

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

[ A={1^k,2^k,3^k,\dots}\subset\mathbb N,\qquad r_k(n):=1_A^{(k)}(n) =|\\{(x_1,\dots,x_k)\in \mathbb N^k:\ x_1^k+\cdots+x_k^k=n\\}|, ]

llm-hunter · gpt pro 5.2 · unverified

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