{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:37:55Z","timestamp":1776724675012,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"222","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n An analogue for composite moduli\n \n \n \n \n m<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n m \\geq 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the Wilson quotient is studied. Various congruences are derived, and the question of when these quotients are divisible by\n \n \n \n m<\/mml:mi>\n m<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is investigated; such an\n \n \n \n m<\/mml:mi>\n m<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n will be called a \u201cWilson number\". It is shown that numbers in certain infinite classes cannot be Wilson numbers. Eight new Wilson numbers up to 500 million were found.\n <\/p>","DOI":"10.1090\/s0025-5718-98-00951-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"843-861","source":"Crossref","is-referenced-by-count":7,"title":["Wilson quotients for composite moduli"],"prefix":"10.1090","volume":"67","author":[{"given":"Takashi","family":"Agoh","sequence":"first","affiliation":[]},{"given":"Karl","family":"Dilcher","sequence":"additional","affiliation":[]},{"given":"Ladislav","family":"Skula","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01153577","article-title":"On Bernoulli and Euler numbers","volume":"61","author":"Agoh, Takashi","year":"1988","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"T. Agoh, K. Dilcher, and L. Skula, Fermat quotients for composite moduli, J. Number Theory 66 (1997), 29\u201350.","DOI":"10.1006\/jnth.1997.2162"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2334-4","volume-title":"Topics in advanced scientific computation","author":"Crandall, Richard E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0387944737"},{"issue":"217","key":"4","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1090\/S0025-5718-97-00791-6","article-title":"A search for Wieferich and Wilson primes","volume":"66","author":"Crandall, Richard","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","volume-title":"History of the theory of numbers. Vol. I: Divisibility and primality","author":"Dickson, Leonard Eugene","year":"1966"},{"key":"6","unstructured":"H. Dubner, Searching for Wilson primes, J. Recreational Math. 21 (1989), 19\u201320."},{"key":"7","unstructured":"R. H. Gonter and E. G. Kundert, All prime numbers up to 18,876,041 have been tested without finding a new Wilson prime, Preprint (1994)."},{"key":"8","doi-asserted-by":"crossref","first-page":"335","DOI":"10.6028\/jres.069B.035","article-title":"Some number-theoretic calculations","volume":"69B","author":"Kloss, K. E.","year":"1965","journal-title":"J. Res. Nat. Bur. Standards Sect. B","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4340","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"M. Lerch, Zur Theorie des Fermatschen Quotienten \\frac{\ud835\udc4e^{\ud835\udc5d-1}-1}\ud835\udc5d=\ud835\udc5e(\ud835\udc4e), Math. Annalen 60 (1905), 471\u2013490.","DOI":"10.1007\/BF01561092"},{"key":"10","doi-asserted-by":"crossref","unstructured":"E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350\u2013360.","DOI":"10.2307\/1968791"},{"key":"11","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9938-4","volume-title":"The book of prime number records","author":"Ribenboim, Paulo","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0387965734"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4330-2","volume-title":"The little book of big primes","author":"Ribenboim, Paulo","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/038797508X"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00951-X\/S0025-5718-98-00951-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00951-X\/S0025-5718-98-00951-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:45:25Z","timestamp":1776721525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00951-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":12,"journal-issue":{"issue":"222","published-print":{"date-parts":[[1998,4]]}},"alternative-id":["S0025-5718-98-00951-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00951-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1998]]}}}