Proof that e squared is Not a Quadratic Irrationality

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This post is based on the paper "Addition a la note sur l'irrationnalité du nombre e" by Joseph Liouville which contains proof of the fact that $e^{2}$ is not a quadratic irrationality.

In previous posts I covered that 1) $e^{2}, e^{4}$ are irrational and 2) $e$ is not a quadratic irrationality. I now present the final chapter in this series namely the:

Another Proof that e squared is Irrational

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In the last post we used the multiplication by $n!$ trick to prove that $e$ is not a quadratic irrationality. In this post we will use same technique albeit in a direct fashion to show that $e^{2}$ is irrational.

Proof that e is Not a Quadratic Irrationality

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Introduction

There are numerous proofs commonly available online for the fact that the Euler's number $e$ is irrational. Then going further we find that $e$ is also a transcendental number which means that it can not be the root of a polynomial equation with integral coefficients and thereby transcends the powers of algebra in a sense. Again the proof that $e$ is transcendental is also available on various places online.

In this post I am going to present the proof that $e$ is not a quadratic irrationality. This is based on the paper "Sur l'irrationnalité du nombre e = 2.718..." by Joseph Liouville.