Znám’s problem

Given a length $k$, is it possible to construct a set of integers $n_1,\mathrm{\dots },n_k$ such that each

$$n_i|(1+\prod _{j\ne i}^n{n}_j)$$

as a proper divisor? This is Znám’s problem.

This problem has solutions for $k>4$, and all solutions for have been found, and a few for higher $k$ are known. The Sylvester sequence provides many of the solutions. At Wayne University in 2001, Brenton and Vasiliu devised an algorithm to exhaustively search for solutions for a given length, and thus they found all solutions for $k=8$. Their algorithm, though smarter than a brute force search, is still computationally intense the larger $k$ gets.

Solutions to the problem have applications in continued fractions and perfectly weighted graphs.

The problem is believed to have been first posed by Štefan Znám (http://planetmath.org/VStefanZnam) in 1972. Qi Sun proved in 1983 that there are solutions for all $k>4$.

References

Brenton, L, and Vasiliu, A. “Znam’s Problem.” Math. Mag. 75, 3-11, 2002.