| Title:
|
Leudesdorf's theorem and Bernoulli numbers (English) |
| Author:
|
Slavutskii, I. Sh. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
35 |
| Issue:
|
4 |
| Year:
|
1999 |
| Pages:
|
299-303 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Wolstenholme-Leudesdorf theorem |
| Keyword:
|
p-integer number |
| Keyword:
|
Bernoulli number |
| Keyword:
|
Wilson quotient |
| Keyword:
|
irregular prime number |
| MSC:
|
11A07 |
| MSC:
|
11B68 |
| idZBL:
|
Zbl 1053.11003 |
| idMR:
|
MR1744517 |
| . |
| Date available:
|
2008-06-06T22:24:38Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107704 |
| . |
| Reference:
|
[1] Agoh, T., Dilcher, K., Skula, L.: Wilson quotients for composite moduli.Comp. Math. 67 (1998). No. 222, 843–861. MR 1464140 |
| Reference:
|
[2] Bayat, M.: A generalization of Wolstenholme’s theorem.Amer. Math. Monthly 109 (1997), 557–560. Zbl 0916.11002, MR 1453658 |
| Reference:
|
[3] Dilcher, K., Skula, L., Slavutskii, I. Sh.: Bernoulli numbers. Bibliography (1713–1990).Queen’s papers in Pure and Applied Mathematics, 1991, No. 87, 175 pp.; Appendix, Preprint (1994), 30 pp. MR 1119305 |
| Reference:
|
[4] Hardy, G. H., Wright, E. M.: An introduction to theory of numbers.5th ed., Oxford Sci. Publ., 1979. MR 0067125 |
| Reference:
|
[5] Lehmer, E.: On congruences involving Bernoulli numbers and quotients of Fermat and Wilson.Ann. Math. 39 (2) (1938), 350–360. MR 1503412 |
| Reference:
|
[6] Leudesdorf, C.: Some results in the elementary theory of numbers.Proc. London Math. Soc. 20 (1889), 199–212. |
| Reference:
|
[7] Rama Rao, M.: An extention of Leudesdorf theorem.J. London Math. Soc. 12 (1937), 247–250. |
| Reference:
|
[8] Slavutskii, I.: Staudt and arithmetic properties on Bernoulli numbers.Hist. Scient. 5 (1995), 70–74. MR 1349737 |
| Reference:
|
[9] Slavutskii, I.: About von Staudt congruences for Bernoulli numbers.to appear. Zbl 1024.11011, MR 1713678 |
| Reference:
|
[10] Washington, L. C.: Introduction to cyclotomic fields.2nd ed., Springer-Verlag, New York, 1997. Zbl 0966.11047, MR 1421575 |
| Reference:
|
[11] Wolstenholme, J.: On certain properties of prime numbers.Quart. J. Math. 5 (1862), 35–39. |
| . |
