{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T16:25:12Z","timestamp":1781108712217,"version":"3.54.1"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"304","license":[{"start":{"date-parts":[[2017,4,26]],"date-time":"2017-04-26T00:00:00Z","timestamp":1493164800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We define a Gauss factorial\n \n \n \n \n \n N<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n !<\/mml:mo>\n <\/mml:mrow>\n N_n!<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to be the product of all positive integers up to\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n that are relatively prime to\n \n \n \n \n n<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n n\\in \\mathbb N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In this paper we study particular aspects of the Gauss factorials\n \n \n \n \n \n \u230a\n \n <\/mml:mo>\n \n \n n<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n M<\/mml:mi>\n <\/mml:mfrac>\n \n \n \u230b\n \n <\/mml:mo>\n n<\/mml:mi>\n <\/mml:msub>\n !<\/mml:mo>\n <\/mml:mrow>\n \\lfloor \\frac {n-1}{M}\\rfloor _n!<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n M<\/mml:mi>\n =<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n M=3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and 6, where the case of\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n having exactly one prime factor of the form\n \n \n \n \n p<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n 6<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n p\\equiv 1\\pmod {6}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is of particular interest. A fundamental role is played by those primes\n \n \n \n \n p<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n 3<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n p\\equiv 1\\pmod {3}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with the property that the order of\n \n \n \n \n \n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:mfrac>\n !<\/mml:mo>\n <\/mml:mrow>\n \\frac {p-1}{3}!<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n modulo\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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 \n \n \n \n n<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n \n \u00b1\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n n\\equiv \\pm 1\\pmod {M}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the above form that satisfy\n \n \n \n \n \n \u230a\n \n <\/mml:mo>\n \n \n n<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n M<\/mml:mi>\n <\/mml:mfrac>\n \n \n \u230b\n \n <\/mml:mo>\n n<\/mml:mi>\n <\/mml:msub>\n !<\/mml:mo>\n \n \u2261\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\lfloor \\frac {n-1}{M}\\rfloor _n!\\equiv 1\\pmod {n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n M<\/mml:mi>\n =<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n M=3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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.\n <\/p>","DOI":"10.1090\/mcom\/3111","type":"journal-article","created":{"date-parts":[[2015,9,23]],"date-time":"2015-09-23T10:12:52Z","timestamp":1443003172000},"page":"899-933","source":"Crossref","is-referenced-by-count":2,"title":["A role for generalized Fermat numbers"],"prefix":"10.1090","volume":"86","author":[{"given":"John","family":"Cosgrave","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karl","family":"Dilcher","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2016,4,26]]},"reference":[{"key":"1","unstructured":"R. Ballinger and W. Keller, List of primes \ud835\udc58.2\u207f+1 for \ud835\udc58<300. 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