Claims

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2025_10_11_fan: Answers a variant (an infinite set of primes), not the Statement (a finite set of at least two primes), so it does not count toward the problem's standing. Steve Fan's thread post proves the infinite case, which erdosproblems.com records as always irrational and which Erdős called a simple exercise.

2026_06_26_fan: Steve Fan's thread post of 26 June 2026 proves that the sum is transcendental, hence irrational, when $P$ has exactly two primes. It factors the sum into two series tied by a rational identity and takes the transcendence of one Hecke–Mahler value from the classical theory. The claim is pending.

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