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Czechoslovak Mathematical Journal

, Volume 62, Issue 1, pp 59–65 | Cite as

A note on the congruence \(\left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right)\) (mod p r )

  • Romeo Meštrović
Article

Abstract

In the paper we discuss the following type congruences:
$$\left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right)(\bmod p^r ),$$
where p is a prime, n, m, k and r are various positive integers with nm ⩾ 1, k ⩾ 1 and r ⩾ 1. Given positive integers k and r, denote by W(k, r) the set of all primes p such that the above congruence holds for every pair of integers nm ⩾ 1. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W(k, r) and inclusion relations between them for various values k and r. In particular, we prove that W(k + i, r) = W(k − 1, r) for all k ⩾ 2, i ⩾ 0 and 3 ⩽ r ⩽ 3k, and W(k, r) = W(1, r) for all 3 ⩽ r ⩽ 6 and k ⩾ 2. We also noticed that some of these properties may be used for computational purposes related to congruences given above.

Keywords

congruence prime powers Lucas’ theorem Wolstenholme prime set W(k, r

MSC 2012

11B65 11A07 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Maritime FacultyUniversity of MontenegroKotorMontenegro

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