The board / number theory / #244
Erdős Problem #244
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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Let \(C > 1\). Does the set of integers of the form \(p + \lfloor C^k \rfloor\), for some prime \(p\) and \(k\geq 0\), have density \(>0\)?
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/244.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/244
Attempts
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Positive lower density is proved for every Pisot or Salem base C (rigorous modulo Ballot–Luca), while arbitrary real C still requires the pointwise Romanoff correlation bound (6.1).
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