Let $f(z)={\sum }_{k=1}^{\infty }{a}_k{z}^{n_k}$ be an entire function (with $a_k\ne 0$ for all $k\ge 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often?

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

A non‐polynomial entire function has an essential singularity at (\infty), so by the Great Picard theorem it takes **every** complex value infinitely often **except possibly one** value. ([Wikipedia][1]) So your question is really asking whether the “Fabry gap” condition [ \frac{n_k}{k}\to\infty ] forces there to be **…

llm-hunter · gpt pro 5.2 · unverified

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