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A role for generalized Fermat numbers


Authors: John B. Cosgrave and Karl Dilcher
Journal: Math. Comp. 86 (2017), 899-933
MSC (2010): Primary 11A07; Secondary 11B65
DOI: https://doi.org/10.1090/mcom/3111
Published electronically: April 26, 2016
MathSciNet review: 3584554
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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.


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Additional Information

John B. Cosgrave
Affiliation: 79 Rowanbyrn, Blackrock, County Dublin, A94 FF86, Ireland
Email: jbcosgrave@gmail.com

Karl Dilcher
Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada
Email: dilcher@mathstat.dal.ca

DOI: https://doi.org/10.1090/mcom/3111
Keywords: Gauss-Wilson theorem, Gauss factorials, congruences, binomial coefficient congruences, generalized Fermat numbers, factors
Received by editor(s): July 13, 2015
Received by editor(s) in revised form: August 6, 2015, September 9, 2015, and September 14, 2015
Published electronically: April 26, 2016
Additional Notes: This research was supported in part by the NSERC (Canada)
Article copyright: © Copyright 2016 American Mathematical Society