# Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial

## TL;DR

This paper proves a novel supercongruence involving basic hypergeometric series modulo the fifth power of a cyclotomic polynomial, extending previous results limited to the fourth power.

## Contribution

It introduces the first proof of a hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial, using the $q$-Zeilberger algorithm.

## Key findings

- Proved a supercongruence modulo the fifth power of a cyclotomic polynomial
- Established related parametric supercongruences
- Extended the known bounds of hypergeometric supercongruences

## Abstract

By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $\Phi_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.