# 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.

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 |