Some functional equations originating from number theory

If the answer is affirmative, the functional equation for homomorphisms is said to be stable in the sense of Hyers and Ulam because the first result concerning the stability of functional equations was presented by Hyers. Indeed, he has answered the question of Ulam for the case where G1 and G2 are assumed to be Banach spaces (see [8]). We may find a number of papers concerning the stability results of various functional equations (see [1,2,3,4,5,6,7,9,10,11,12,13,14,15,16] and the references cited therein). According to a well-known theorem in number theory, a positive integer of the form m2n, where each divisor of n is not a square of the integer, can be represented as a sum of two squares of integer if and only if every prime factor of n is not of the form 4k + 3. In the proof of this theorem, we make use of the following elementary equalities