Year: 2025
Type: preprint
Abstract: In this paper we are gonna domonstrate a simple proof of the irrationality of e q in a more direct manner, using infinite series. 1 Background In 1737, Euler gave the first proof of irrationality of e by using the simple continued fraction expansion see [1] [2], Then Fourier introduce a really nice proof by contradiction [3][4], years after Liouville adapted Fourier's methods to prove that e 2 is also irrational, in fact, e can't be the root of a second degree equation with rational coefficients, and by using the same trick, he managed to prove the irrationality of e 4 [5], But e 3 , e 5 seems to be difficult to obtain using the previous ideas, In 1891, Hurwitz explained how it is possible to prove that e is not root of a third degree [6]. However, in [7], Hermite uses a new idea and some calculus to prove the strong result that e q is irrational for any non-zero rational q. But the question remains Can Fourier's proof of the irrationality of e be modified to establish the irrationality of e q , where q is an integer?
Author: Ali Chtatbi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
OpenAlex ID: W4413926529