Znám’s problem

Given a length k, is it possible to construct a set of integers n1,,nk such that each


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

This problem has solutions for k>4, and all solutions for 4<k<9 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 fractionsDlmfMathworldPlanetmath 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.


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

Title Znám’s problem
Canonical name ZnamsProblem
Date of creation 2013-03-22 15:47:39
Last modified on 2013-03-22 15:47:39
Owner Mravinci (12996)
Last modified by Mravinci (12996)
Numerical id 8
Author Mravinci (12996)
Entry type Definition
Classification msc 11A55
Synonym Znam’s problem