On irregular prime power divisors of the Bernoulli numbers

Kellner, Bernd

doi:10.1090/S0025-5718-06-01887-4

On irregular prime power divisors of the Bernoulli numbers
HTML articles powered by AMS MathViewer

by Bernd C. Kellner PDF

Math. Comp. 76 (2007), 405-441 Request permission

Abstract:

Let $B_n$ ($n = 0, 1, 2, \ldots$) denote the usual $n$th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not \equiv B_{l+p-1}/(l+p-1) \operatorname {mod}{p^2}$. We show that for every $r \geq 1$ the congruence $B_{m_r}/m_r \equiv 0 \operatorname {mod}{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \operatorname {mod}{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $\chi _{(p, l)}$ which is a zero of a certain $p$-adic zeta function $\zeta _{p, l}$ originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) $p$-adic expansion of $\chi _{(p, l)}$ for irregular pairs $(p,l)$ with $p$ below 1000.

References

J. C. Adams, Table of the values of the first sixty-two numbers of Bernoulli, J. Reine Angew. Math. 85 (1878), 269–272.
Joe Buhler, Richard Crandall, Reijo Ernvall, Tauno Metsänkylä, and M. Amin Shokrollahi, Irregular primes and cyclotomic invariants to 12 million, J. Symbolic Comput. 31 (2001), no. 1-2, 89–96. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806208, DOI 10.1006/jsco.1999.1011
L. Carlitz, Some theorems on Kummer’s congruences, Duke Math. J. 20 (1953), 423–431. MR 56009
L. Carlitz, Note on irregular primes, Proc. Amer. Math. Soc. 5 (1954), 329–331. MR 61124, DOI 10.1090/S0002-9939-1954-0061124-6
L. Carlitz, Kummer’s congruence for the Bernoulli numbers, Portugal. Math. 19 (1960), 203–210. MR 125054
T. Clausen, Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351–352.
Jean Fresnel, Nombres de Bernoulli et fonctions $L$ $p$-adiques, Ann. Inst. Fourier (Grenoble) 17 (1967), no. fasc. 2, 281–333 (1968) (French). MR 224570
Ralph Greenberg, Iwasawa theory—past and present, Class field theory—its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 335–385. MR 1846466, DOI 10.2969/aspm/03010335
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
Wells Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653–657. MR 347727, DOI 10.1090/S0025-5718-1974-0347727-0
B. C. Kellner, Über irreguläre Paare höherer Ordnungen , Diplomarbeit, Mathematisches Institut der Georg–August–Universität zu Göttingen, Germany, 2002. (Also available at http://www.bernoulli.org/$\sim$bk/irrpairord.pdf)
Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003, DOI 10.1007/978-1-4612-1112-9
Tomio Kubota and Heinrich-Wolfgang Leopoldt, Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen, J. Reine Angew. Math. 214(215) (1964), 328–339 (German). MR 163900
E. E. Kummer, Allgemeiner Beweis des Fermat’schen Satzes, dass die Gleichung $x^{\lambda }+y^{\lambda }=z^{\lambda }$ durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten $\lambda$, welche ungerade Primzahlen sind und in den Zählern der ersten $({\lambda }-3)/2$ Bernoulli’schen Zahlen als Factoren nicht vorkommen, J. Reine Angew. Math. 40 (1850), 130–138.
E. E. Kummer, Über eine allgemeine Eigenschaft der rationalen Entwicklungscoefficienten einer bestimmten Gattung analytischer Functionen, J. Reine Angew. Math. 41 (1851), 368–372.
F. Pollaczek, Über die irregulären Kreiskörper der $l$-ten und $l^2$-ten Einheitswurzeln, Math. Z. 21 (1924), 1–38. JFM 50.0111.02
Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1964 (1964), 51–57 (German). MR 163899
H. S. Vandiver, On Bernoulli’s numbers and Fermat’s last theorem, Duke Math. J. 3 (1937), no. 4, 569–584. MR 1546011, DOI 10.1215/S0012-7094-37-00345-4
K. G. C. von Staudt, Beweis eines Lehrsatzes die Bernoulli’schen Zahlen betreffend, J. Reine Angew. Math. 21 (1840), 372–374.
Samuel S. Wagstaff Jr., The irregular primes to $125000$, Math. Comp. 32 (1978), no. 142, 583–591. MR 491465, DOI 10.1090/S0025-5718-1978-0491465-4
Samuel S. Wagstaff Jr., Prime divisors of the Bernoulli and Euler numbers, Number theory for the millennium, III (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 357–374. MR 1956285
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Itaru Yamaguchi, On a Bernoulli numbers conjecture, J. Reine Angew. Math. 288 (1976), 168–175. MR 424669, DOI 10.1515/crll.1976.288.168