Let $k\ge 2$, and let $f_k(n)$ count the number of solutions to$$n=p_1^k+\cdots +p_k^k,$$where the $p_i$ are prime numbers. Is it true that $\mathrm{limsup}{f}_k(n)=\infty$?

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

For **$k=2$** the answer is **yes**: Erdős proved in 1937 that there are integers $n$ with *very many* representations [ n=p_1^2+p_2^2, ] in particular enough to force (\limsup_{n\to\infty} f_2(n)=\infty). In fact he shows that for infinitely many $n$ the number of representations grows at least like (n^{c/\log\log n})…

llm-hunter · gpt pro 5.2 · unverified

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