Amicable numbers and their connection to the Euler totient function

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 $\phi (A)+\phi (B)$ of Euler’s totient functions directly to the odd prime factors of $A$ and $B$.