Leudesdorf's theorem and Bernoulli numbers

I. Sh. Slavutsky

Archivum Mathematicum (1999)

  • Volume: 035, Issue: 4, page 299-303
  • ISSN: 0044-8753

Abstract

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For m ∈ , ( m , 6 ) = 1 , it is proved the relations between the sums W ( m , s ) = ∑ i = 1 , ( i , m ) = 1 m - 1 i - s , s ∈ , 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.

How to cite

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Slavutsky, I. Sh.. "Leudesdorf's theorem and Bernoulli numbers." Archivum Mathematicum 035.4 (1999): 299-303. <http://eudml.org/doc/248357>.

@article{Slavutsky1999,
abstract = {For $m\in $, $(m,6)=1$, it is proved the relations between the sums \[ W(m,s)=\sum \_\{i=1, (i,m)=1\}^\{m-1\} i^\{-s\}\,, \quad \quad s\in \,, \] 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.},
author = {Slavutsky, I. Sh.},
journal = {Archivum Mathematicum},
keywords = {Wolstenholme-Leudesdorf theorem; p-integer number; Bernoulli number; Wilson quotient; irregular prime number; Wolstenholme-Leudesdorf theorem; Bernoulli number; Wilson quotient; irregular prime},
language = {eng},
number = {4},
pages = {299-303},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Leudesdorf's theorem and Bernoulli numbers},
url = {http://eudml.org/doc/248357},
volume = {035},
year = {1999},
}

TY - JOUR
AU - Slavutsky, I. Sh.
TI - Leudesdorf's theorem and Bernoulli numbers
JO - Archivum Mathematicum
PY - 1999
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 035
IS - 4
SP - 299
EP - 303
AB - For $m\in $, $(m,6)=1$, it is proved the relations between the sums \[ W(m,s)=\sum _{i=1, (i,m)=1}^{m-1} i^{-s}\,, \quad \quad s\in \,, \] 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.
LA - eng
KW - Wolstenholme-Leudesdorf theorem; p-integer number; Bernoulli number; Wilson quotient; irregular prime number; Wolstenholme-Leudesdorf theorem; Bernoulli number; Wilson quotient; irregular prime
UR - http://eudml.org/doc/248357
ER -

References

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  8. Staudt and arithmetic properties on Bernoulli numbers, Hist. Scient. 5 (1995), 70–74. MR1349737
  9. About von Staudt congruences for Bernoulli numbers, to appear. Zbl1024.11011MR1713678
  10. Introduction to cyclotomic fields, 2nd ed., Springer-Verlag, New York, 1997. Zbl0966.11047MR1421575
  11. On certain properties of prime numbers, Quart. J. Math. 5 (1862), 35–39. 

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