Linear Independence Of Some Irrational Numbers

TL;DR

This paper introduces an analytic method to prove the linear independence over rationals of specific small sets of real numbers, including key mathematical constants and their powers.

Contribution

It provides a new analytic technique for establishing linear independence of certain subsets of irrational numbers, simplifying proofs for well-known constants.

Findings

Proves the linear independence of {1, e, π} over rationals.

Establishes independence of {1, e, π^{-1}}.

Shows independence of {1, π^r, π^s} for fixed integers r<s.

Abstract

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples $\{1, e, \pi\}$, $\{1, e, \pi^{-1}\}$, and $\{1, \pi^r, \pi^s\}$, where $1\leq r<s $ are fixed integers.