On a functional equation arising from number theory

On functional equations of the multiplicative type

“…If G is a commutative semigroup without unity 1, and T is a commutative semigroup and S is a commutative group, then the assertion of Theorem 2.1 still holds. For a proof see [4].…”

“…3) or there are other solutions. This question was addressed by Chávez and Sahoo [4] (and subsequently fixed an error in [4] by Chung and Sahoo [5]) in the following theorem. One of the well known properties of matrices is the following: The determinant of the product of two square matrices is the product of their determinants.…”

“…If one lets the quadratic form on the right hand side of the Proth identity x 4 + y 4 + (x + y) 4 = 2 x 2 + xy + y 2 2 (1.1) be f (x, y) = x 2 + xy + y 2 (1.2) then it can be easily verified that f (ux − vy, uy + v(x + y)) = f (x, y) f (u, v) (1.3) for all x, y, u, v ∈ R. It is obvious that f given in (1.2) is a solution of the above functional equation (1.3). To prove the above mentioned result, Blecksmith and Broudno [3] used the Proth identity and the above functional equation (1.3).…”

On a Functional Equation Arising From Proth Identity

“…As an example, suppose m = 4, then one can find an integer n, namely n = 1729 such that 2 · 1729 2 can be expressed as the sum of three fourth powers in four ways: About the integer 1729, once Ramanujan said to Hardy that the number 1729 is an interesting number because it is the smallest number expressible as the sum of two cubes in two different ways (namely 9 3 + 10 3 = 1 3 + 12 3 = 1729). In [3], the authors determined the general solution f : R 2 → R of the functional equation (1.3) (see Theorem 2.4 in [3]). In the general solution, a factor was missing from the solution.…”

“…The interested reader should refer to the books [5] and [4] on the subject of functional equations and stabilities. In this paper, first we determine the general solutions f : R 2 → R of the functional equation (1.3) for all x, y, u, v ∈ R by a simple but different method than the one used in [3] and correct the solution in Theorem 2.4 of [3]. Then, by finding a condition (see (3.2)) for the solution f of the following functional inequality to be unbounded we investigate both the bounded and unbounded solutions of the functional inequality…”