JP Journal of Algebra, Number Theory and Applications
Volume 42, Issue 2, Pages 159 - 169
(May 2019) http://dx.doi.org/10.17654/NT042020159 |
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ON A GENERALIZATION OF A LUCAS’ RESULT ON THE PASCAL’S TRIANGLE AND MERSENNE NUMBERS
Atsushi Yamagami and Hideyuki Harada
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Abstract: 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. |
Keywords and phrases: the (k-)Pascal’s triangle, p-adic repunits, Mersenne numbers.
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