Amicable numbers and their connection to the Euler totient function

TL;DR

This paper explores the history of amicable numbers and introduces a new characterization for pairs with a GCD that is a power of two, linking their prime factorizations to Euler's totient function.

Contribution

It presents a novel approach to characterize amicable pairs with GCD as a power of two using their prime factorizations and totient function identities.

Findings

Explicit symmetric identities relating totient sums to prime factors

New characterization for amicable pairs with GCD as a power of two

Connections between prime factorization patterns and amicability

Abstract

A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest common divisor is a power of two, using their distinct prime factorizations. Specifically, we examine pairs of the forms $A=2^n ab, B=2^n cd$, $A=2^n abc, B=2^n de$, and $A=2^n abc, B=2^n def$. From these configurations, we establish explicit symmetric identities that relate the sum $\varphi(A)+\varphi(B)$ of Euler's totient functions directly to the odd prime factors of $A$ and $B$.