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Some Congruences Involving Binomial Coefficient and Fermat Quotient

Wei-Wei Qi

TL;DR

The paper addresses high-order congruences for sums involving d^{-k} binomial(x, k) binomial(x+k, k) divided by binomial(2k, k) with p-adic x and Fermat quotients. It introduces a key summation identity and uses binomial algebra, harmonic-number relations, and Wolstenholme-type results to obtain p-adic congruences modulo p^4 (and p^5 in special cases), culminating in Theorems 1.1, 1.3, and 1.5. Corollaries are derived by specializing x to particular values such as -1/2, -1/3, -1/4, and -1/6, and by applying auxiliary harmonic-number congruences. Overall, the work extends the landscape of binomial-coefficient congruences in the p-adic setting and provides explicit, higher-order congruence formulas useful for number theory and combinatorics.

Abstract

In this paper, we investigate some congruences involving sums of $\frac{d^{-k}{x\choose k}{x+k\choose k}}{2k \choose k}$, where $x$ be a $p$-adic integer, $k$ be a non-negative integer, and $d$ $(d\neq 0)$ be a rational number.

Some Congruences Involving Binomial Coefficient and Fermat Quotient

TL;DR

The paper addresses high-order congruences for sums involving d^{-k} binomial(x, k) binomial(x+k, k) divided by binomial(2k, k) with p-adic x and Fermat quotients. It introduces a key summation identity and uses binomial algebra, harmonic-number relations, and Wolstenholme-type results to obtain p-adic congruences modulo p^4 (and p^5 in special cases), culminating in Theorems 1.1, 1.3, and 1.5. Corollaries are derived by specializing x to particular values such as -1/2, -1/3, -1/4, and -1/6, and by applying auxiliary harmonic-number congruences. Overall, the work extends the landscape of binomial-coefficient congruences in the p-adic setting and provides explicit, higher-order congruence formulas useful for number theory and combinatorics.

Abstract

In this paper, we investigate some congruences involving sums of , where be a -adic integer, be a non-negative integer, and be a rational number.

Paper Structure

This paper contains 3 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Proofs of the Theorems
  3. Proofs of the corollaries

Key Result

Theorem 1.1

Let $p\geq 3$ be a prime, $d$ ($d\neq 0$) be a rational number, $x$ be a $p$-adic integer and $m:=(x-\langle x\rangle_p)/p$, $\langle x\rangle_p \in \{0,1,2,\dots p-1\}$. Then modulo $p^4$

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 3.1