{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:40:40Z","timestamp":1776786040969,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"257","license":[{"start":{"date-parts":[[2007,8,1]],"date-time":"2007-08-01T00:00:00Z","timestamp":1185926400000},"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 Let\n \n \n \n \n B<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n B_n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (\n \n \n \n \n n<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n <\/mml:mrow>\n n = 0, 1, 2, \\ldots<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) denote the usual\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n th Bernoulli number. Let\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a positive even integer where\n \n \n \n \n l<\/mml:mi>\n =<\/mml:mo>\n 12<\/mml:mn>\n <\/mml:mrow>\n l=12<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n l<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 16<\/mml:mn>\n <\/mml:mrow>\n l \\geq 16<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is well known that the numerator of the reduced quotient\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n B<\/mml:mi>\n l<\/mml:mi>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n l<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n |B_l\/l|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a product of powers of irregular primes. Let\n \n \n \n \n (<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n l<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (p,l)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an irregular pair with\n \n \n \n \n \n B<\/mml:mi>\n l<\/mml:mi>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n l<\/mml:mi>\n \u2262<\/mml:mo>\n \n B<\/mml:mi>\n \n l<\/mml:mi>\n +<\/mml:mo>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n l<\/mml:mi>\n +<\/mml:mo>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n mod<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \n p<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mrow>\n B_l\/l \\not \\equiv B_{l+p-1}\/(l+p-1) \\operatorname {mod}{p^2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We show that for every\n \n \n \n \n r<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n r \\geq 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the congruence\n \n \n \n \n \n B<\/mml:mi>\n \n \n m<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n m<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \u2261\n \n <\/mml:mo>\n 0<\/mml:mn>\n mod<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \n p<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mrow>\n B_{m_r}\/m_r \\equiv 0 \\operatorname {mod}{p^r}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has a unique solution\n \n \n \n \n m<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n m_r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n where\n \n \n \n \n \n m<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \u2261\n \n <\/mml:mo>\n l<\/mml:mi>\n mod<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mrow>\n m_r \\equiv l \\operatorname {mod}{p-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n l<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n m<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n ><\/mml:mo>\n (<\/mml:mo>\n p<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \n p<\/mml:mi>\n \n r<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n l \\leq m_r > (p-1)p^{r-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The sequence\n \n \n \n \n (<\/mml:mo>\n \n m<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n )<\/mml:mo>\n \n r<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n (m_r)_{r \\geq 1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n defines a\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -adic integer\n \n \n \n \n \n \u03c7\n \n <\/mml:mi>\n \n (<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n \n l<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \\chi _{(p,\\,l)}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which is a zero of a certain\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -adic zeta function\n \n \n \n \n \n \u03b6\n \n <\/mml:mi>\n \n p<\/mml:mi>\n ,<\/mml:mo>\n \n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\zeta _{p,\\,l}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n originally defined by T.\u00a0Kubota and H.\u00a0W.\u00a0Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated)\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -adic expansion of\n \n \n \n \n \n \u03c7\n \n <\/mml:mi>\n \n (<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n \n l<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msub>\n \\chi _{(p,\\,l)}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for irregular pairs\n \n \n \n \n (<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n l<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (p,l)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n below 1000.\n <\/p>","DOI":"10.1090\/s0025-5718-06-01887-4","type":"journal-article","created":{"date-parts":[[2006,11,1]],"date-time":"2006-11-01T16:50:02Z","timestamp":1162399802000},"page":"405-441","source":"Crossref","is-referenced-by-count":7,"title":["On irregular prime power divisors of the Bernoulli numbers"],"prefix":"10.1090","volume":"76","author":[{"given":"Bernd","family":"Kellner","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,8,1]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"J. C. Adams, Table of the values of the first sixty-two numbers of Bernoulli, J. Reine Angew. Math. 85 (1878), 269\u2013272.","DOI":"10.1515\/crll.1878.85.269"},{"issue":"1-2","key":"2","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1006\/jsco.1999.1011","article-title":"Irregular primes and cyclotomic invariants to 12 million","volume":"31","author":"Buhler, Joe","year":"2001","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1215\/S0012-7094-53-02043-2","article-title":"Some theorems on Kummer\u2019s congruences","volume":"20","author":"Carlitz, L.","year":"1953","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"329","DOI":"10.2307\/2032249","article-title":"Note on irregular primes","volume":"5","author":"Carlitz, L.","year":"1954","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"5","first-page":"203","article-title":"Kummer\u2019s congruence for the Bernoulli numbers","volume":"19","author":"Carlitz, L.","year":"1960","journal-title":"Portugal. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0032-5155","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"T. Clausen, Lehrsatz aus einer Abhandlung \u00fcber die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351\u2013352.","DOI":"10.1002\/asna.18400172204"},{"key":"7","doi-asserted-by":"crossref","first-page":"281","DOI":"10.5802\/aif.271","article-title":"Nombres de Bernoulli et fonctions \ud835\udc3f \ud835\udc5d-adiques","volume":"17","author":"Fresnel, Jean","year":"1967","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"335","DOI":"10.2969\/aspm\/03010335","article-title":"Iwasawa theory\u2014past and present","author":"Greenberg, Ralph","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/4931469116"},{"key":"9","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2103-4","volume-title":"A classical introduction to modern number theory","volume":"84","author":"Ireland, Kenneth","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/038797329X","edition":"2"},{"key":"10","doi-asserted-by":"publisher","first-page":"653","DOI":"10.2307\/2005943","article-title":"Irregular prime divisors of the Bernoulli numbers","volume":"28","author":"Johnson, Wells","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","unstructured":"B. C. Kellner, \u00dcber irregul\u00e4re Paare h\u00f6herer Ordnungen, Diplomarbeit, Mathematisches Institut der Georg\u2013August\u2013Universit\u00e4t zu G\u00f6ttingen, Germany, 2002. (Also available at http:\/\/www.bernoulli.org\/\u223cbk\/irrpairord.pdf)"},{"key":"12","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1112-9","volume-title":"$p$-adic numbers, $p$-adic analysis, and zeta-functions","volume":"58","author":"Koblitz, Neal","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0387960171","edition":"2"},{"key":"13","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1515\/crll.1964.214-215.328","article-title":"Eine \ud835\udc5d-adische Theorie der Zetawerte. I. Einf\u00fchrung der \ud835\udc5d-adischen Dirichletschen \ud835\udc3f-Funktionen","volume":"214(215)","author":"Kubota, Tomio","year":"1964","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"E. E. Kummer, Allgemeiner Beweis des Fermat\u2019schen Satzes, dass die Gleichung \ud835\udc65^{\ud835\udf06}+\ud835\udc66^{\ud835\udf06}=\ud835\udc67^{\ud835\udf06} durch ganze Zahlen unl\u00f6sbar ist, f\u00fcr alle diejenigen Potenz-Exponenten \ud835\udf06, welche ungerade Primzahlen sind und in den Z\u00e4hlern der ersten (\ud835\udf06-3)\/2 Bernoulli\u2019schen Zahlen als Factoren nicht vorkommen, J. Reine Angew. Math. 40 (1850), 130\u2013138.","DOI":"10.1515\/crll.1850.40.130"},{"key":"15","doi-asserted-by":"crossref","unstructured":"E. E. Kummer, \u00dcber eine allgemeine Eigenschaft der rationalen Entwicklungscoefficienten einer bestimmten Gattung analytischer Functionen, J. Reine Angew. Math. 41 (1851), 368\u2013372.","DOI":"10.1515\/crll.1851.41.368"},{"key":"16","doi-asserted-by":"crossref","unstructured":"F. Pollaczek, \u00dcber die irregul\u00e4ren Kreisk\u00f6rper der \ud835\udc59-ten und \ud835\udc59\u00b2-ten Einheitswurzeln, Math. Z. 21 (1924), 1\u201338. JFM 50.0111.02","DOI":"10.1007\/BF01187449"},{"key":"17","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3254-2","volume-title":"A course in $p$-adic analysis","volume":"198","author":"Robert, Alain M.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387986693"},{"key":"18","first-page":"51","article-title":"Zu zwei Bemerkungen Kummers","volume":"1964","author":"Siegel, Carl Ludwig","year":"1964","journal-title":"Nachr. Akad. Wiss. G\\\"{o}ttingen Math.-Phys. Kl. II","ISSN":"https:\/\/id.crossref.org\/issn\/0065-5295","issn-type":"print"},{"issue":"4","key":"19","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1215\/S0012-7094-37-00345-4","article-title":"On Bernoulli\u2019s numbers and Fermat\u2019s last theorem","volume":"3","author":"Vandiver, H. S.","year":"1937","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"20","doi-asserted-by":"crossref","unstructured":"K. G. C. von Staudt, Beweis eines Lehrsatzes die Bernoulli\u2019schen Zahlen betreffend, J. Reine Angew. Math. 21 (1840), 372\u2013374.","DOI":"10.1515\/crll.1840.21.372"},{"issue":"142","key":"21","doi-asserted-by":"publisher","first-page":"583","DOI":"10.2307\/2006167","article-title":"The irregular primes to 125000","volume":"32","author":"Wagstaff, Samuel S., Jr.","year":"1978","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"22","isbn-type":"print","first-page":"357","article-title":"Prime divisors of the Bernoulli and Euler numbers","author":"Wagstaff, Samuel S., Jr.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/1568811527"},{"key":"23","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1934-7","volume-title":"Introduction to cyclotomic fields","volume":"83","author":"Washington, Lawrence C.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0387947620","edition":"2"},{"key":"24","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1515\/crll.1976.288.168","article-title":"On a Bernoulli numbers conjecture","volume":"288","author":"Yamaguchi, Itaru","year":"1976","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2007-76-257\/S0025-5718-06-01887-4\/S0025-5718-06-01887-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-257\/S0025-5718-06-01887-4\/S0025-5718-06-01887-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:46:42Z","timestamp":1776782802000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-257\/S0025-5718-06-01887-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,8,1]]},"references-count":24,"journal-issue":{"issue":"257","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0025-5718-06-01887-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01887-4","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":[[2006,8,1]]}}}