# On some conjectural congruences

## TL;DR

This paper proves several conjectured congruences involving prime numbers and binomial coefficients, confirming recent hypotheses in number theory.

## Contribution

It verifies conjectured congruences proposed by Guo and Schlosser, advancing understanding of prime-related binomial sum congruences.

## Key findings

- Confirmed specific congruences modulo p^5 for primes p>3.
- Validated conjectures involving binomial coefficients and prime moduli.
- Enhanced the theoretical framework for congruences in number theory.

## Abstract

In this paper, we confirm some congruences conjectured by V.J.W. Guo and M.J. Schlosser recently. For example, we show that for primes $p>3$, $$ \sum_{k=0}^{p-1}(2pk-2k-1)\frac{\left(\frac{-1}{p-1}\right)_k^{2p-2}}{(k!)^{2p-2}}\equiv0\pmod{p^5}. $$