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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  2. A generalization of Wolstenholme’s theorem, Amer. Math. Monthly 109 (1997), 557–560. Zbl0916.11002MR1453658
  3. Bernoulli numbers. Bibliography (1713–1990), Queen’s papers in Pure and Applied Mathematics, 1991, No. 87, 175 pp.; Appendix, Preprint (1994), 30 pp. MR1119305
  4. An introduction to theory of numbers, 5th ed., Oxford Sci. Publ., 1979. MR0067125
  5. On congruences involving Bernoulli numbers and quotients of Fermat and Wilson, Ann. Math. 39 (2) (1938), 350–360. MR1503412
  6. Some results in the elementary theory of numbers, Proc. London Math. Soc. 20 (1889), 199–212. 
  7. An extention of Leudesdorf theorem, J. London Math. Soc. 12 (1937), 247–250. 
  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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