The Odd Solutions of Equations Involving Euler-Like Function

Abstract

φ e ( n ) is a function similar to Euler function φ ( n ). We discussed and obtained all the odd solutions of the equations φ e ( xy ) = φ e ( x ) + 2 φ e ( y) , φ e ( xy ) = 2 φ e ( x ) + 3 φ e ( y) and φ e ( xyz ) = φ e ( x ) + φ e ( y) + φ e ( z) by using the methods and techniques of elementary number theory.