A problem posed by the Slovak mathematician Stefan Znám in 1972 asking whether, for all integers , there exist integers all greater than 1 such that is a proper divisor of for each . The answer is negative for (Jának and Skula 1978) and affirmative for (Sun Qi 1983). Sun Qi also gave a lower bound for the number of solutions.

All solutions for have now been computed, summarized in the table below. The numbers of solutions for , 3, ... terms are 0, 0, 0, 2, 5, 15, 93, ... (OEIS A075441), and the solutions themselves are given by OEIS A075461.

known solutions | references | ||

2 | 0 | -- | Jának and Skula (1978) |

3 | 0 | -- | Jának and Skula (1978) |

4 | 0 | -- | Jának and Skula (1978) |

5 | 2 | 2, 3, 7, 47, 395 | |

2, 3, 11, 23, 31 | |||

6 | 5 | 2, 3, 7, 43, 1823, 193667 | |

2, 3, 7, 47, 403, 19403 | |||

2, 3, 7, 47, 415, 8111 | |||

2, 3, 7, 47, 583, 1223 | |||

2, 3, 7, 55, 179, 24323 | |||

7 | 15 | 2, 3, 7, 43, 1807, 3263447, 2130014000915 | Jának and Skula (1978) |

2, 3, 7, 43, 1807, 3263591, 71480133827 | Cao, Liu, and Zhang (1987) | ||

2, 3, 7, 43, 1807, 3264187, 14298637519 | |||

2, 3, 7, 43, 3559, 3667, 33816127 | |||

2, 3, 7, 47, 395, 779831, 6020372531 | |||

2, 3, 7, 67, 187, 283, 334651 | |||

2, 3, 11, 17, 101, 149, 3109 | |||

2, 3, 11, 23, 31, 47063, 442938131 | |||

2, 3, 11, 23, 31, 47095, 59897203 | |||

2, 3, 11, 23, 31, 47131, 30382063 | |||

2, 3, 11, 23, 31, 47243, 12017087 | |||

2, 3, 11, 23, 31, 47423, 6114059 | |||

2, 3, 11, 23, 31, 49759, 866923 | |||

2, 3, 11, 23, 31, 60563, 211031 | |||

2, 3, 11, 31, 35, 67, 369067 | |||

8 | 93 | Brenton and Vasiliu (1998) | |

9 | ? | 2, 3, 7, 43, 1807, 3263443, | Sun (1983) |

10650056950807, | |||

113423713055421844361000447, | |||

2572987736655734348107429290411162753668127385839515 | |||

10 | ? | 2, 3, 11, 23, 31, 47059, | Sun (1983) |

2214502423, 4904020979258368507, | |||

24049421765006207593444550012151040547, | |||

115674937446230858658157460659985774139375256845351399814552547262816571295 |

Cao and Sun (1988) showed that and Cao and Jing (1998) that there are solutions for . A solution for was found by Girgensohn in 1996: 3, 4, 5, 7, 29, 41, 67, 89701, 230865947737, 5726348063558735709083, followed by large numbers having 45, 87, and 172 digits.

It has been observed that all known solutions to Znám's problem provide a decomposition of 1 as an Egyptian fraction

Conversely, every solution to this Diophantine equation is a solution to Znám's problem, unless for some .