Access the full text.
Sign up today, get DeepDyve free for 14 days.
M. Eie (1996)
A Note on Bernoulli Numbers and Shintani Generalized Bernoulli PolynomialsTransactions of the American Mathematical Society, 348
L. Washington (1982)
Introduction to Cyclotomic Fields
K. Iwasawa (1972)
Lectures on p-adic L-functions
Jean-Pierre Serre (1973)
A Course in Arithmetic
Kinkichi Iwasawa (1972)
Lectures on p-Adic L-functions (AM-74)
N. Koblitz (1980)
P-adic analysis : a short course on recent work
Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB n(n = 1,2,3,...) assert that (1-p m-1)B m/m isp-integral and $$(1 - p^{m - 1} )\frac{{B_m }}{m} \equiv (1 - p^{n - 1} )\frac{{B_n }}{n}(\bmod p^{N + 1} )$$ ifm ≡ n (mod (p-1)p n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN, $$\frac{1}{m}\{ B_m (\frac{\alpha }{k}) - p^{m - 1} B_m (\frac{\beta }{k})\} \equiv \frac{1}{n}\{ B_n (\frac{\alpha }{k}) - p^{n - 1} B_n (\frac{\beta }{k})\} (\bmod p^{N + 1} )$$ ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p n). HereB n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences for Bernoulli polynomials of odd indices.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Aug 27, 2008
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.