Some Congruences Involving Binomial Coefficient and Fermat Quotient

TL;DR

This paper explores new congruences involving binomial coefficients and Fermat quotients, focusing on sums with p-adic integers and rational parameters, advancing understanding in number theory.

Contribution

It introduces novel congruences related to binomial sums and Fermat quotients involving p-adic integers and rational parameters.

Findings

Derived new congruences involving binomial sums and Fermat quotients.

Extended existing theories to include p-adic and rational parameters.

Provided proofs and formulations for the congruences studied.

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.