ZnamsProblem 2006-03-21 16:23:55 2006-10-17 16:31:23 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a length $k$, is it possible to construct a set of integers $n_1, \ldots , n_k$ such that each $$n_i |(1 + \prod_{j \ne i}^n n_j)$$ as a proper divisor? This is \emph{Zn\'am'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 \PMlinkescapetext{State} 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 \PMlinkname{\v{S}tefan Zn\'am}{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. {\bf 75}, 3-11, 2002.