We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.

We will study the solution of a congruence,x ≡g^1/2)ωg(2ⁿ) mod 2ⁿ, depending on the integersg andn, where ω_g(2ⁿ) denotes the order ofg modulo 2ⁿ. Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation ¦〈f(x), f(y)〉¦ = ¦〈x,y〉¦. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD_n be a suitable subset of ℝⁿ. If a function f:D_n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉¦ ¦〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D_n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D_n.