A role for generalized Fermat numbers
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.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.identifier.uri | http://hdl.handle.net/10222/71449 | |
dc.publisher | American Mathematical Society | en_US |
dc.relation.ispartof | Mathematics of Computation | en_US |
dc.rights.license | Creative Commons Attribution - Non-Commercial - No Derivatives (CC BY-NC-ND) | |
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 |
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