| Summary: | The Pascal’s triangle is generalized to “the <i>k</i>-Pascal’s triangle” with any integer <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. Let <i>p</i> be any prime number. In this article, we prove that for any positive integers <i>n</i> and <i>e</i>, the <i>n</i>-th row in the <inline-formula> <math display="inline"> <semantics> <msup> <mi>p</mi> <mi>e</mi> </msup> </semantics> </math> </inline-formula>-Pascal’s triangle consists of integers which are congruent to 1 modulo <i>p</i> if and only if <i>n</i> is of the form <inline-formula> <math display="inline"> <semantics> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow> <mi>e</mi> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>p</mi> <mi>e</mi> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </semantics> </math> </inline-formula> with some integer <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. This is a generalization of a Lucas’ result asserting that the <i>n</i>-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i>n</i> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>p</mi> <mi>e</mi> </msup> </msup> <mo>≡</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>p</mi> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>p</mi> <mrow> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </msup> <mrow> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="4pt"></mspace> <msup> <mi>p</mi> <mi>e</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of binomial expansions which we could prove for any prime number <i>p</i> and any positive integer <i>e</i>. 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.
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