On a Generalization of a Lucas' Result and an Application to the 4-Pascal's Triangle

The Pascal's triangle is generalized to "the k-Pascal's triangle" with any integer k >= 2. Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the pe-Pascal's triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form pem-1pe-1 with some integer m >= 1. This is a generalization of a Lucas' result asserting that the n-th row in the (2-)Pascal's triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal's triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence (x+1)pe equivalent to(xp+1)pe-1(modpe) of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue "Number Theory and Symmetry," since we prove a symmetric property on the 4-Pascal's triangle by means of a number-theoretical property of binomial expansions.