A note on Apery's constant is transcendental

Abstract

Beuker's [2] considers the following integral $$ \int_{0}^{1}\int_{0}^{1} \frac{-\log xy}{1-xy} P_n(x)P_n(y)\ dx dy$$If $d_n=\text{LCM}(1,2,...,n)$, then $$ 0<\frac{|A_n+B_n\zeta(3)|}{d_n^3}<2(\sqrt{2}-1)^{4n} \zeta(3) $$ for some $A_n,B_n\in\mathbb{Z}$. We establish that if Apery's constant is algebraic then the above inequality fails to be true. This proves that $\zeta(3)$ is