# Znám’s problem

Given a length $k$, is it possible to construct a set of integers $n_{1},\ldots,n_{k}$ such that each

 $n_{i}|(1+\prod_{j\neq i}^{n}n_{j})$

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

This problem has solutions for $k>4$, and all solutions for $4 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.

Title Znám’s problem ZnamsProblem 2013-03-22 15:47:39 2013-03-22 15:47:39 Mravinci (12996) Mravinci (12996) 8 Mravinci (12996) Definition msc 11A55 Znam’s problem