The \n p\n -Adic Valuations of Sums of Binomial Coefficients

Abstract

<jats:p>In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M2 >\n <mi>p</mi>\n </math>\n </jats:inline-formula> be an odd prime and let <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M3 >\n <mi>h</mi>\n <mo>∈</mo>\n <mi>ℤ</mi>\n </math>\n </jats:inline-formula> with <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M4 >\n <mn>2</mn>\n <mi>h</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>≡</mo>\n <mn>0</mn>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mi mathvariant= normal >mod</mi>\n <mi>p</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>. For <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M5 >\n <mi>a</mi>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>ℤ</mi>\n </mrow>\n <mrow>\n <mo>+</mo>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M6 >\n <msup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mi>a</mi>\n </mrow>\n </msup>\n <mo>></mo>\n <mn>3</mn>\n </math>\n </jats:inline-formula>, we show that <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M7 >\n <mstyle displaystyle= true >\n <msubsup>\n <mo stretchy= false >∑</mo>\n <mrow>\n <mi>k</mi>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mi>a</mi>\n </mrow>\n </msup>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msubsup>\n <mrow>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mtable>\n <mtr columnalign= left >\n <mtd columnalign= left >\n <mrow>\n <mi>h</mi>\n <msup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mi>a</mi>\n </mrow>\n </msup>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mtd>\n </mtr>\n <mtr columnalign= left >\n <mtd columnalign= left >\n <mi>k</mi>\n </mtd>\n </mtr>\n </mtable>\n </mrow>\n </mfenced>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mtable>\n <mtr columnalign= left >\n <mtd columnalign= left >\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n </mtd>\n </mtr>\n <mtr columnalign= left >\n <mtd columnalign= left >\n <mi>k</mi>\n </mtd>\n </mtr>\n </mtable>\n </mrow>\n </mfenced>\n <msup>\n <mrow>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mo>−</mo>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mrow>\n <mi>h</mi>\n <mo>/</mo>\n <mn>2</mn>\n </mrow>\n </mrow>\n </mfenced>\n </mrow>\n </mfenced>\n </mrow>\n <mrow>\n <mi>k</mi>\n </mrow>\n </msup>\n </mrow>\n </mstyle>\n <mo>≡</mo>\n <mn>0</mn>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mi mathvariant= normal >mod</mi>\n <msup>\n <mrow>\n <mi>p</mi>\n </mrow>\n <mrow>\n <mi>a</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>. Also, for any <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M8 >\n <mi>n</mi>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>ℤ</mi>\n </mrow>\n <mrow>\n <mo>+</mo>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula>, we have <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M9 >\n <msub>\n <mrow>\n <mi>ν</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msub>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mstyle displaystyle= true >\n <msubsup>\n <mrow>\n <mo stretchy= false >∑</mo>\n </mrow>\n <mrow>\n <mi>k</mi>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msubsup>\n <mrow>\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <mtable>\n