### Abstract

Wendt's determinant of order n is the circulant determinant W_{n} whose (i,j)-th entry is the binomial coefficient ( _{i-j}^{n} , for 1≤i, j≤n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then W_{pk} ≡ 1 (mod p^{k}) and W_{npk} ≡ W_{n} (mod p). If q is another prime, distinct from p, and h any positive integer, then W_{phqk} ≡ W_{phqk} (mod pq). Furthermore, if p is odd, then W_{p} ≡ 1 + p ((_{p-1}^{2p-1}) - 1) (mod p^{5}). In particular, if p ≥ 5, then W_{p} ≡ 1 (mod p^{4}). Also, if m and n are relatively prime positive integers, then W_{m} W_{n} divides W_{mn}.

Original language | English (US) |
---|---|

Pages (from-to) | 45-57 |

Number of pages | 13 |

Journal | Journal of Number Theory |

Volume | 115 |

Issue number | 1 |

DOIs | |

State | Published - Nov 1 2005 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Fingerprint Dive into the research topics of 'Arithmetical properties of wendt's determinant'. Together they form a unique fingerprint.

## Cite this

*Journal of Number Theory*,

*115*(1), 45-57. https://doi.org/10.1016/j.jnt.2004.09.003