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

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 n ⩾ m ⩾ 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 n ⩾ m ⩾ 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.