Are there infinitely many primary pseudoperfect numbers: 1/p_1+…+1/p_k = 1 − 1/m? (Erdős #313)
Statement
Are there infinitely many solutions to $$\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},$$ where $m\ge 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?
Acceptance. FULLY RESOLVES: a proof that infinitely many such solutions exist (equivalently, infinitely many primary pseudoperfect numbers), or a proof that there are only finitely many; machine-checkable preferred, else a full written proof. ADVANCES (each independently checkable): (a) discover a new primary pseudoperfect number beyond the $8$ known, giving $m$, its full prime factorisation, and a verification that $\sum_{p\mid m}\tfrac1p+\tfrac1m=1$ (equivalently the explicit prime unit-fraction identity) — a fully machine-checkable witness extending A054377; (b) extend the exhaustive search bound below which the known $8$ are proven complete, with the search program and an exhaustiveness certificate; (c) a conditional infinitude or finiteness result under a clearly stated hypothesis. Deliver the new number + verification, the search code + attained bound, or the (conditional) proof.
Background
Posed by Erdős and Graham [ErGr80, p.40]; listed as open on erdosproblems.com/313 (fetched 2026-07-21, status 'open'). Small solutions include $\tfrac12+\tfrac13=1-\tfrac16$ and $\tfrac12+\tfrac13+\tfrac17=1-\tfrac1{42}$. Clearing denominators forces $m=p_1\cdots p_k$, so there is at most one solution for each $m$, and the admissible $m$ are exactly the primary pseudoperfect numbers — integers $m>1$ with $\sum_{p\mid m}\tfrac1p+\tfrac1m=1$. Only $8$ are known (OEIS A054377: $2,\,6,\,42,\,1806,\,47058,\,2214502422,\,52495396602,\,8490421583559688410706771261086$); whether infinitely many exist is the content of the problem. Attacker's tool: an exhaustive / structured search for a ninth primary pseudoperfect number (building candidate squarefree $m$ prime-by-prime under the reciprocal-sum constraint with aggressive pruning, in the style of Sylvester–Znám / perfect-number searches), extending A054377 and its verified completeness bound; a positive resolution instead requires a genuine proof of infinitude.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #313 (T. F. Bloom) | website |
| REF-02 | OEIS A054377 — primary pseudoperfect numbers | website |
| REF-03 | Primary pseudoperfect number (Wikipedia) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.