OPEN
This is open, and cannot be resolved with a finite computation.
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?
For example,\[\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}\]and\[\frac{1}{2}+\frac{1}{3}+\frac{1}{7}=1-\frac{1}{42}.\]It is clear that we must have $m=p_1\cdots p_k$, and hence in particular there is at most one solution for each $m$. The integers $m$ for which there is such a solution are known as
primary pseudoperfect numbers, and there are $8$ known, listed in
A054377 at the OEIS.
View the LaTeX source
Additional thanks to: Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #313, https://www.erdosproblems.com/313, accessed 2026-01-16
