dc.contributor.author | Cosgrave, John B. | |

dc.contributor.author | Dilcher, Karl | |

dc.date.accessioned | 2016-04-19T18:16:21Z | |

dc.date.available | 2016-04-19T18:16:21Z | |

dc.date.issued | 2016 | |

dc.identifier.uri | http://hdl.handle.net/10222/71449 | |

dc.description | Post-print version of the article, issued prior to publication. | |

dc.description.abstract | We define a Gauss factorial $N_n!$ to be the product of all positive integers
up to $N$ that are relatively prime to $n\in\mathbb N$. In this paper we study
particular aspects of the Gauss factorials $\lfloor\frac{n-1}{M}\rfloor_n!$
for $M=3$ and 6, where the case of $n$ having exactly one prime factor
of the form $p\equiv 1\pmod{6}$ is of particular interest. A fundamental
role is played by those primes $p\equiv 1\pmod{3}$ with the property that the
order of $\frac{p-1}{3}!$ modulo $p$ is a power of 2 or 3 times a power of 2;
we call them Jacobi primes. Our main results are characterizations of those
$n\equiv\pm 1\pmod{M}$ of the above form that satisfy
$\lfloor\frac{n-1}{M}\rfloor_n!\equiv 1\pmod{n}$, $M=3$ or 6, in terms
of Jacobi primes and certain prime factors of generalized Fermat numbers. We
also describe the substantial and varied computations used for this paper. | en_US |

dc.description.sponsorship | Natural Sciences and Engineering Research Council of Canada | en_US |

dc.publisher | American Mathematical Society | en_US |

dc.relation.ispartof | Mathematics of Computation | en_US |

dc.subject | Gauss-Wilson theorem | en_US |

dc.subject | Gauss factorials | en_US |

dc.subject | congruences | en_US |

dc.subject | binomial coefficient congruences | en_US |

dc.subject | generalized Fermat numbers | en_US |

dc.subject | Factors (Algebra) | en_US |

dc.title | A role for generalized Fermat numbers | en_US |

dc.type | Article | en_US |

dc.rights.license | Creative Commons Attribution - Non-Commercial - No Derivatives (CC BY-NC-ND) | |