Erdős Problem #517
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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If `f(z) = ∑ aₖzⁿₖ` is an entire function (with `aₖ ≠ 0` for all `k`) such that `nₖ / k → ∞`, is it true that `f` assumes every value infinitely often?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/517.Prizenone
Topicsanalysis
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/517
Attempts
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Explicitly constructed and verified an infinite-order, non-Fejér Fabry-gap entire function that assumes every value infinitely often; the universal problem reduces exactly to excluding zero-density Ta
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