The main application of the method of auxiliary polynomials is in diophantine approximation and transcendence. But before these topics comes irrationality: one seeks to prove that a given number is not in Q. One of the earliest examples is of course
It is proved in any elementary text on number theory that e is irrational; the proof is based on the rapid convergence of the series together with the reasonable behaviour of the denominators. We give a proof nevertheless.
Consider the truncation
for n = 0, 1, 2, · · ·. The first term ...