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Is
irrational? (Here is the th prime.)
Source: erdosproblems.com/251
No claim settles this problem.
Open, as the site's label (OPEN) also has it. No unconditional proof or disproof is known. Two claimed conditional proofs, both under Kuperberg's uniform Hardy–Littlewood prime-tuples conjecture, have claim pages, Land and Ringer; neither is refereed or independently reviewed, and both would leave the problem open even if accepted.