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A role for generalized Fermat numbers

dc.contributor.authorCosgrave, John B.
dc.contributor.authorDilcher, Karl
dc.date.accessioned2016-04-19T18:16:21Z
dc.date.available2016-04-19T18:16:21Z
dc.date.issued2016
dc.descriptionPost-print version of the article, issued prior to publication.
dc.description.abstractWe 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.sponsorshipNatural Sciences and Engineering Research Council of Canadaen_US
dc.identifier.urihttp://hdl.handle.net/10222/71449
dc.publisherAmerican Mathematical Societyen_US
dc.relation.ispartofMathematics of Computationen_US
dc.rights.licenseCreative Commons Attribution - Non-Commercial - No Derivatives (CC BY-NC-ND)
dc.subjectGauss-Wilson theoremen_US
dc.subjectGauss factorialsen_US
dc.subjectcongruencesen_US
dc.subjectbinomial coefficient congruencesen_US
dc.subjectgeneralized Fermat numbersen_US
dc.subjectFactors (Algebra)en_US
dc.titleA role for generalized Fermat numbersen_US
dc.typeArticleen_US

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