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 en_US 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. 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)
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