# Nonexistence of $D(4)$-quintuples

## TL;DR

This paper proves that $D(4)$-quintuples do not exist, resolving a long-standing conjecture using a combination of classical and novel mathematical techniques, and introduces a new version of Rickert's theorem applicable to certain $D(4)$-quadruples.

## Contribution

It provides a proof of the nonexistence of $D(4)$-quintuples and presents a new version of Rickert's theorem for specific $D(4)$-quadruples.

## Key findings

- $D(4)$-quintuples do not exist.
- A new version of Rickert's theorem is introduced.
- The proof combines classical and innovative methods.

## Abstract

In this paper we prove a conjecture that $D(4)$-quintuple does not exist using both classical and new methods. Also, we give a new version of the Rickert's theorem that can be applied on some $D(4)$-quadruples.