On the Transcendence of π
📜 Abstract
A simple proof of the transcendence of π is given, using elementary properties of determinants and symmetric polynomials. It is similar to, but somewhat simpler than, the original Lindemann proof. The proof involves consideration of algebraic symmetric polynomials and the implication of a vanishing determinant.
✨ Summary
The paper “On the Transcendence of π” by Morris Marden provides a simplified proof of the transcendence of π, a mathematical constant that is irrational and non-algebraic. The paper uses elementary properties of determinants and symmetric polynomials to offer a proof that is claimed to be simpler than the original Lindemann proof. Although this work revisits and simplifies an existing mathematical proof, it does not seem to have had a significant influence on subsequent research or industry applications, as evidenced by a lack of citations or discussions in more recent publications. The Lindemann-Weierstrass theorem, which the original proof of π’s transcendence is part of, remains foundational in mathematics, particularly in transcendental number theory. However, there is no direct evidence found indicating specific influence or further applications resulting from Marden’s paper itself.