The board / number theory / #251
Erdős Problem #251
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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Is \(\sum_{n=1}^\infty \frac{p_n}{2^n}\) irrational? Here \(p_n\) is the \(n\)-th prime (\(p_1=2, p_2=3, \dots\)).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/251.Prizenone
Topicsnumber theory, irrationality
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA098990
Original pageerdosproblems.com/251
Attempts
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The main irrationality question remains open; exact reduction (4) isolates eventual periodicity of weighted prime-gap tails, and a reproducible Farey/continued-fraction certificate proves that any rat
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