Applications of Bernoulli Identities to Kummer-type Congruences
by
Minking Eie
National Chung Cheng University
Coauthors: Yao Lin Ong (National Chung Cheng University)

Through simple identities among a particular type of rational functions, we are able to produce identities among Bernoulli numbers and then congruent relations

(1-pm-1)  Bm m \equiv  1 mpN+1 å [( 1 <= j < pN+1) || (j, p)=1]  jm -  1 2 å [( 1 <= j < pN+1) || (j, p)=1]  jm-1 mod pN+1

when the odd prime p has the property that p-1 is not a divisor of the positive even integer m. With such relations, we are able to produce new identities among Bernoulli numbers as well as reprove Kummer-type congruences such as

r å l=0  æ è r l ö ø (-1)r-l  Bm+\omega l m+\omega l \equiv 0 mod (per, pm-1)

when \omega is a multiple of (p-1)p^e-1, e >= 1.

http://www.math.ccu.edu.tw/teacher/mkeie/mkeie.htm

Date received: March 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-61.