Problem 313

Are there infinitely many solutions to

\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},

where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?

Source: erdosproblems.com/313

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Open. Eleven solutions are known (OEIS A054377, revision of 30 September 2026): the eight the site counts; $N_9=5998279018951962402$ and $N_{10}=N_9(N_9+1)$ , with nine and ten prime factors, published in Wang's arXiv preprint of May 2026; and a second solution with ten prime factors, $2318487344461212044808266715505249967391337588451045543300838$ , in an OEIS comment of Pedro Martins of 27 September 2026. The entry also reports no further term below $10^{24}$ (12 August 2026); the equations were recomputed here. Infinitude is unproved: Wang's Theorem 19.5 gives it only under an unproved prime-points hypothesis of Bateman--Horn type, recorded on Wang's conditional claim page (2026). No proof, disproof or proof claim for the exact statement was found in the search whose scope the Current assessment records; this is a bounded negative finding.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 40. Library home: Erdős and Graham 1980.
Wa26: Wang, H., Port fillings for primary pseudoperfect numbers. arXiv:2605.21518v1 (18 May 2026), 23 pages; an unrefereed preprint. Library home: Wang 2026; result pages for Theorems 9.1, 11.1 and 19.5.
BJM00: Butske, W., Jaje, L. M. and Mayernik, D. R., On the equation $\sum_{p\mid N}1/p+1/N=1$ , pseudoperfect numbers, and perfectly weighted graphs. Math. Comp. 69 (2000), 407--420, DOI 10.1090/S0025-5718-99-01088-1; cited by Wang, and linked from the OEIS entry, for the computation that for each $r\le8$ there is exactly one primary pseudoperfect number with $r$ prime factors. Not held; second-hand here.
OEIS: Sequence A054377, Primary pseudoperfect numbers, The On-Line Encyclopedia of Integer Sequences (revision 198, 30 September 2026): the data line $2,6,42,1806,47058,2214502422,52495396602,5998279018951962402$ , with the three longer terms in comments.

Problem 313

Status

Proof claims

Discussion