{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:52:36Z","timestamp":1776786756548,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"260","license":[{"start":{"date-parts":[[2008,4,17]],"date-time":"2008-04-17T00:00:00Z","timestamp":1208390400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n A prime\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is called a\n Fibonacci-Wieferich prime<\/italic>\n if\n \n \n \n \n \n F<\/mml:mi>\n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n (<\/mml:mo>\n \n \n p<\/mml:mi>\n 5<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2261\n \n <\/mml:mo>\n 0<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n \n p<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n F_{p-({p\\over 5})}\\equiv 0\\pmod {p^2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n F<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n F_n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n th Fibonacci number. We report that there exist no such primes\n \n \n \n \n p<\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n \n \u00d7\n \n <\/mml:mo>\n \n 10<\/mml:mn>\n \n 14<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n p>2\\times 10^{14}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . A prime\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is called a\n Wolstenholme prime<\/italic>\n if\n \n \n \n \n \n \n \n \n (<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n \n 2<\/mml:mn>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n <\/mml:mrow>\n <\/mml:mrow>\n \n \u2261\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n \n p<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n {2p-1\\choose p-1}\\equiv 1\\pmod {p^4}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes\n \n \n \n \n p<\/mml:mi>\n ><\/mml:mo>\n \n 10<\/mml:mn>\n 9<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n p>10^9<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Wolstenholme, in 1862, proved that\n \n \n \n \n \n \n \n \n (<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n \n 2<\/mml:mn>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n <\/mml:mrow>\n <\/mml:mrow>\n \n \u2261\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n \n p<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n {2p-1\\choose p-1}\\equiv 1\\pmod {p^3}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for all primes\n \n \n \n \n p<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n p\\ge 5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is estimated by a heuristic argument that the \u201cprobability\u201d that\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is Fibonacci-Wieferich (independently: that\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is Wolstenholme) is about\n \n \n \n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:mrow>\n 1\/p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient\n \n \n \n \n \n F<\/mml:mi>\n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n (<\/mml:mo>\n \n \n p<\/mml:mi>\n 5<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:mrow>\n F_{p-({p\\over 5})}\/p<\/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 .\n <\/p>","DOI":"10.1090\/s0025-5718-07-01955-2","type":"journal-article","created":{"date-parts":[[2007,7,26]],"date-time":"2007-07-26T07:46:03Z","timestamp":1185435963000},"page":"2087-2094","source":"Crossref","is-referenced-by-count":41,"title":["A search for Fibonacci-Wieferich and Wolstenholme primes"],"prefix":"10.1090","volume":"76","author":[{"given":"Richard","family":"McIntosh","sequence":"first","affiliation":[]},{"given":"Eric","family":"Roettger","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2007,4,17]]},"reference":[{"issue":"203","key":"1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.2307\/2152942","article-title":"Irregular primes and cyclotomic invariants to four million","volume":"61","author":"Buhler, J.","year":"1993","journal-title":"Math. 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Dickson, The History of the Theory of Numbers, vol. 1, Reprinted: Chelsea Publishing Company, New York, 1966."},{"key":"6","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9307-8","volume-title":"Fermat's last theorem","volume":"50","author":"Edwards, Harold M.","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902309"},{"key":"7","series-title":"Problem Books in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-26677-0","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387208607","edition":"3"},{"key":"8","doi-asserted-by":"publisher","first-page":"113","DOI":"10.2307\/2005468","article-title":"Irregular primes and cyclotomic invariants","volume":"29","author":"Johnson, Wells","year":"1975","journal-title":"Math. 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Sun, private communication, 2005."},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"371","DOI":"10.4064\/aa-60-4-371-388","article-title":"Fibonacci numbers and Fermat\u2019s last theorem","volume":"60","author":"Sun, Zhi Hong","year":"1992","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"18","doi-asserted-by":"publisher","first-page":"525","DOI":"10.2307\/2309169","article-title":"Fibonacci series modulo \ud835\udc5a","volume":"67","author":"Wall, D. D.","year":"1960","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/0898-1221(82)90026-8","article-title":"The influence of computers in the development of number theory","volume":"8","author":"Williams, H. C.","year":"1982","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"20","unstructured":"J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math 5 (1862) 35\u201339."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01955-2\/S0025-5718-07-01955-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01955-2\/S0025-5718-07-01955-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:03:38Z","timestamp":1776783818000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-260\/S0025-5718-07-01955-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,4,17]]},"references-count":20,"journal-issue":{"issue":"260","published-print":{"date-parts":[[2007,10]]}},"alternative-id":["S0025-5718-07-01955-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-07-01955-2","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":[[2007,4,17]]}}}