## Using Legendre Polynomials in irrationality proofs and establishing irrationality measures

The Legendre polynomials are a set of polynomial solutions to the Legendre’s Differential equation:

$(1-x^{2})y^{''}-2xy^{'}+n(n+1)y=0$

The edifice of most irrationality arguments use the fact the repeated integration by parts can be used on generating functions involving the polynomial.Another useful property is that these polynomials have integer coefficients.

Now,I’ll present a few important examples which display this property.

## Refreshing problem

There isn’t too much going in this question.Nonetheless,I just liked it for some bloody reason.

• Let $a$ and $n$ be positive integers where $a>1$.If $a^{n}+1$ is prime,prove that $a$ is even and n is of the form $2^{m}$ where $m \in \mathbb{N}$

## The irrationality of e

This is the post excerpt.

The irrationality of e can be proved using the infinite series expansion.

$e=\sum_{k=0}^{\infty} \frac{1}{k!}$

Assume that e is rational.i.e $e=\frac{p}{q}$ where $p \in \mathbb{Z}$ and $q \in \mathbb{Z}-{0}$.