znams_problem

Znám's Problem: For every $k \geq 2$, does there exist a set of $k$ positive integers $\{x_1, \ldots, x_k\}$ such that each $x_i$ divides $1 + \prod_{j \neq i} x_j$?

Štefan Znám posed this in 1975. Solutions are known for $k = 2, 3, \ldots, 9$. Example for $k=2$: $\{2, 3\}$ since $2 \mid 1+3=4$ and $3 \mid 1+2=3$. The question is whether solutions exist for all $k$. Closely related to Egyptian fraction representations and Sylvester sequences.

Source: Znám, Š. (1975). J. Number Theory 7, 303–310. Also: Sun, Z.W. (2009). On Znám's problem and generalized Znám's problem. Integers 9, A44.