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

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 $n\ge 1$, 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 $n\ge b$, 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.

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