A Golden Pair of Identities in the Theory of Numbers

TL;DR

This paper uncovers new mathematical identities linking the golden ratio, Moebius function, Euler totient function, and natural logarithm, revealing unexpected connections in number theory.

Contribution

It introduces novel identities involving these fundamental number theory functions and the golden ratio, highlighting previously unnoticed relationships and inverse properties.

Findings

Derived identities involving the golden ratio and number theory functions

Established a connection between the golden ratio and integer factorization

Revealed an inverse relationship between Moebius and Euler totient functions

Abstract

We find an interesting relationship between the golden ratio, the Moebius function, the Euler totient function and the natural logarithm - central players in the theory of numbers. A number of identities involving the golden ratio and its reciprocal are proved, including an expression for the base of the natural logarithm. The theorem and corollaries highlight a connection between the golden ratio and the factorization of integers that is not obvious; and display a sort of inverse relationship between the Moebius function and Euler totient function.