A note on the Irrationality of $\zeta(5)$ and Higher Odd Zeta Values

TL;DR

This paper proves the irrationality of  and extends the method to demonstrate the irrationality of all higher odd zeta values, using contradiction, Diophantine equations, and induction.

Contribution

It introduces a novel proof technique for irrationality of odd zeta values, generalizing previous methods with a systematic approach.

Findings

Proves  is irrational.

Extends the proof to all higher odd zeta values.

Uses Diophantine equations and induction in the proof.

Abstract

In this note, we prove the irrationality of $\zeta(5)$ and generalize the method to prove the irrationality of all higher odd zeta values. Our proof relies on the method of contradiction, existence of solution of a system of Linear Diophantine equation, and mathematical induction. For $n\geq 1$, we denote $d_n=\text{lcm}(1,2,...,n)$. In the first part of the article, we assume $\zeta(5)$ is rational, say $a/b$. We observe that for $n\geq b$, there exists a system of equations involving linear combination of $\zeta(5)$ that has a solution. Later using the existence of solution of the Linear Diophantine equation, we show that such a system of linear combination of $\zeta(5)$ has no solution, which is a contradiction. In the second part of the article, we generalise this method for all higher odd zeta values.