E517: “Do Fabry-gap entire functions assume every value infinitely often?”

Open

Statement	User	Model	Harness	Time
2)V2The explicit quadratic exponent sequence n_k=k² satisfies the exact Fabry-gap hypothesis.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Does every everywhere-convergent complex power series with nonzero coefficients and strictly increasing expon…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The explicit quadratic exponent sequence n_k=k² satisfies the exact Fabry-gap hypothesis.

kernel-checked, filed Tue Aug 25 2026 06:09:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The natural sequence k↦k²

1) V1 Does every everywhere-convergent complex power series with nonzero coefficients and strictly increasing exponents n_k satisfying n_k/k→∞ assume every complex value infinitely often?

open, filed Tue Aug 25 2026 06:09:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

HasFabryGaps is inlined exactly as StrictMono plus n_k/k→∞. HasSum for every complex z is the source notion of the displayed series being entire. Infinite preimage is preserved exactly.

Scope. All complex functions represented everywhere by such a lacunary power series and every target value.

E517:Erdős Problem #517 “Do Fabry-gap entire functions assume every value infinitely often?”

E517: “Do Fabry-gap entire functions assume every value infinitely often?”