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Keywords:
Wolstenholme-Leudesdorf theorem; p-integer number; Bernoulli number; Wilson quotient; irregular prime number
Wolstenholme-Leudesdorf theorem; p-integer number; Bernoulli number; Wilson quotient; irregular prime number
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.
References:
[1] Agoh, T., Dilcher, K., Skula, L.: Wilson quotients for composite moduli. Comp. Math. 67 (1998). No. 222, 843–861. MR 1464140
[2] Bayat, M.: A generalization of Wolstenholme’s theorem. Amer. Math. Monthly 109 (1997), 557–560. MR 1453658 | Zbl 0916.11002
[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
[4] Hardy, G. H., Wright, E. M.: An introduction to theory of numbers. 5th ed., Oxford Sci. Publ., 1979. MR 0067125
[5] Lehmer, E.: On congruences involving Bernoulli numbers and quotients of Fermat and Wilson. Ann. Math. 39 (2) (1938), 350–360. MR 1503412
[6] Leudesdorf, C.: Some results in the elementary theory of numbers. Proc. London Math. Soc. 20 (1889), 199–212.
[7] Rama Rao, M.: An extention of Leudesdorf theorem. J. London Math. Soc. 12 (1937), 247–250.
[8] Slavutskii, I.: Staudt and arithmetic properties on Bernoulli numbers. Hist. Scient. 5 (1995), 70–74. MR 1349737
[9] Slavutskii, I.: About von Staudt congruences for Bernoulli numbers. to appear. MR 1713678 | Zbl 1024.11011
[10] Washington, L. C.: Introduction to cyclotomic fields. 2nd ed., Springer-Verlag, New York, 1997. MR 1421575 | Zbl 0966.11047
[11] Wolstenholme, J.: On certain properties of prime numbers. Quart. J. Math. 5 (1862), 35–39.
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