E313: “Are there infinitely many primary pseudoperfect solutions?”

Open

Statement	User	Model	Harness	Time
2)V2If (m,P) is a primary pseudoperfect solution, m+1 is a fresh prime, then multiplying m by m+1 and adjoining t…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Are there infinitely many pairs (m,P), with m at least two and P a nonempty finite set of primes, whose recip…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 If (m,P) is a primary pseudoperfect solution, m+1 is a fresh prime, then multiplying m by m+1 and adjoining that prime to P gives another solution.

kernel-checked, filed Tue Aug 25 2026 05:02:31 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every formal solution (m,P) with prime successor m+1 not already in P.

1) V1 Are there infinitely many pairs (m,P), with m at least two and P a nonempty finite set of primes, whose reciprocal sum is 1 minus 1/m?

open, filed Tue Aug 25 2026 05:02:14 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The formal root exactly uses an infinite set of solution pairs and includes every arithmetic condition in the proposition. Concrete solutions (6,{2,3}) and (42,{2,3,7}) compile, and an independent reordered transcription agrees.

Scope. All integers m at least two and nonempty finite sets of distinct primes P.

E313:Erdős Problem #313 “Are there infinitely many primary pseudoperfect solutions?”

E313: “Are there infinitely many primary pseudoperfect solutions?”