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LEUDESDORF'S THEOREM AND BERNOULLI NUMBERS

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*I. Sh. Slavutskii*

**Address.** St. Hamarva, 4, P.O.Box 23393, Akko, ISRAEL
**E-mail:** nick1@luckynet.co.il

**Abstract.** For $m\in \N$, $(m,6)=1$, it is proved the relations
between the sums $$ W(m,s)=\sum_{i=1, (i,m)=1}^{m-1} i^{-s}\,, \quad \quad
s\in \N\,, $$ and Bernoulli numbers. The result supplements the known theorems
of C. Leudesdorf, N. Rama Rao and others. As the application it is obtained
some connections between the sums $W(m,s)$ and Agoh's functions, Wilson
quotients, the indices irregularity of Bernoulli numbers.

**AMSclassification.** Primary 11A07; Secondary 11B68

**Keywords.** Wolstenholme-Leudesdorf theorem, $p$-integer number,
Bernoulli number, Wilson quotient, irregular prime number