A Golden Pair of Identities in the Theory of Numbers

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.