In [2, Section 1.4], the Pascal’s triangle is generalized to “the k‑Pascal’s triangle” with any integer k ≥ 2. Let n be a positive integer and p be any prime number. In this article, we prove that the n-th row in the p-Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form  with some integer  This is a generalization of a Lucas’ result [1, Example I in Section 228] asserting that the n-th row in the Pascal’s triangle consists of odd integers if and only if n is a Mersenne number.

the (k-)Pascal’s triangle, p-adic repunits, Mersenne numbers.