{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:43:35Z","timestamp":1776789815415,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,10,31]],"date-time":"2009-10-31T00:00:00Z","timestamp":1256947200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n With the aid of a computer network we have performed a search for primes\n \n \n \n \n p<\/mml:mi>\n ><\/mml:mo>\n \n 10<\/mml:mn>\n \n 12<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n p>10^{12}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and revealed a new prime\n \n \n \n \n p<\/mml:mi>\n =<\/mml:mo>\n 6692367337<\/mml:mn>\n <\/mml:mrow>\n p=6692367337<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for which its least primitive root\n \n \n \n \n (<\/mml:mo>\n \n mod<\/mml:mtext>\n <\/mml:mrow>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (\\textrm {mod} p)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and its least primitive root\n \n \n \n \n (<\/mml:mo>\n \n mod<\/mml:mtext>\n <\/mml:mrow>\n \n p<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n (\\textrm {mod} p^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are not equal.\n <\/p>","DOI":"10.1090\/s0025-5718-08-02090-5","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"1193-1195","source":"Crossref","is-referenced-by-count":8,"title":["A new prime \ud835\udc5d for which the least primitive root (\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc5d) and the least primitive root (\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc5d\u00b2) are not equal"],"prefix":"10.1090","volume":"78","author":[{"given":"A.","family":"Paszkiewicz","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,10,31]]},"reference":[{"key":"1","unstructured":"N. Beeger, On a new case of the congruence 2^{\ud835\udc5d-1}\u22611(\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc5d\u00b2). Messenger of Mathematics, 51 (1922), pp. 149\u2013150."},{"issue":"217","key":"2","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"},{"issue":"250","key":"3","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1090\/S0025-5718-04-01666-7","article-title":"Solutions of the congruence \ud835\udc4e^{\ud835\udc5d-1}\u22611 (mod \ud835\udc5d^{\ud835\udc5f})","volume":"74","author":"Keller, Wilfrid","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"251","key":"4","doi-asserted-by":"publisher","first-page":"1559","DOI":"10.1090\/S0025-5718-05-01723-0","article-title":"The continuing search for Wieferich primes","volume":"74","author":"Knauer, Joshua","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","first-page":"106","article-title":"Primitive roots for the first million primes and their powers","author":"Litver, E. L.","year":"1971"},{"key":"6","unstructured":"W. Meissner, \u00dcber die Teilbarkeit von 2^{\ud835\udc5d}-2 durch das Quadrat der Primzahl \ud835\udc5d=1093, Sitzungsberichte Preuss. Akad. Wiss. (1913), pp. 663\u2013667."},{"issue":"203","key":"7","doi-asserted-by":"publisher","first-page":"361","DOI":"10.2307\/2152960","article-title":"New solutions of \ud835\udc4e^{\ud835\udc5d-1}\u22611 (mod \ud835\udc5d\u00b2)","volume":"61","author":"Montgomery, Peter L.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"240","key":"8","doi-asserted-by":"publisher","first-page":"1781","DOI":"10.1090\/S0025-5718-01-01382-5","article-title":"Numerical calculation of the density of prime numbers with a given least primitive root","volume":"71","author":"Paszkiewicz, A.","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02090-5\/S0025-5718-08-02090-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02090-5\/S0025-5718-08-02090-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:58:58Z","timestamp":1776787138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02090-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,10,31]]},"references-count":8,"journal-issue":{"issue":"266","published-print":{"date-parts":[[2009,4]]}},"alternative-id":["S0025-5718-08-02090-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02090-5","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":[[2008,10,31]]}}}