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Let be an entire function which is not a monomial. Let count the number of with such that . (This is a finite quantity if is not a monomial.)
Is it possible for
Is it possible for
Source: erdosproblems.com/1117
A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.
The site labels the problem OPEN (page last edited 29 December 2025). The site's commentary records that the first question has the answer yes, by Herzog and Piranian (1968), a pending partial claim on their page, and that the second question is open, with an approximate affirmative analogue by Glücksam and Pardo-Simón [GlPa24]. The site's proof-claims tab carries a claim credited to Qiyuan Gu, submitted 2026-09-05 and listed there as a full claim; the claim's notes say that GPT-6 Astra generated the proofs and that GPT-5.6 Sol and Claude Opus 5 were used for editorial review. It asserts outside a countable set of radii, where is the gap between the exponents of the first two nonzero terms of , so that for every non-monomial entire , a negative answer to the second question. It is pending on its page as a partial claim settling the second question (proof-claims thread accessed 2026-10-06). The derived standing, claimed with the value answered, departs from OPEN because a pending claim answers the second question, which the site's commentary records as open, and with the pending affirmative answer to the first every part is covered by a pending claim.