## 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

## 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.## References

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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
- 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: https://doi.org/10.1090/mcom/3111
- MathSciNet review: 3584554