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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A role for generalized Fermat numbers
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by John B. Cosgrave and Karl Dilcher PDF
Math. Comp. 86 (2017), 899-933 Request permission


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:
  • Karl Dilcher
  • Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada
  • Email:
  • 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)
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 899-933
  • MSC (2010): Primary 11A07; Secondary 11B65
  • DOI:
  • MathSciNet review: 3584554