Annales Univ. Sci. Budapest., Sect. Comp. 25 (2006) 65-77 ADDITIVE UNIQUENESS SETS FOR MULTIPLICATIVE FUNCTIONS K.-H. Indlekofer (Paderborn, Germany) Bui Minh Phong (Budapest, Hungary) Abstract. We proved that if a multiplicative function F and a positive integer k satisfy F (2) =0, F (5) =1 and F ( n 2 + m 2 + k +1 ) = F (n 2 + k)+ F (m 2 + 1) for all n, m ∈ IN, then F (n)= n for all positive integers n, (n, 2) = 1. 1. Introduction In this paper, let IN and P stand for the set of positive integers and prime numbers, respectively. We denote by M the set of all multiplicative functions f such that f (1) = 1. Furthermore, we deal with the set B of non-negative integers which can be represented as a sum of two squares of integers and with S the set of all squares of positive integers. In the following subsets A and B of IN are called additive uniqueness sets (AU-sets) for M if f ∈M satisfies f (a + b)= f (a)+ f (b) for all a ∈ A and b ∈ B, then f (n)= n for all n ∈ IN . In 1992 C.Spiro [6] showed that A = B = P are AU-sets for M. In [1] it is proved that A = S and B = P are also AU-sets for The research was supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and the Hungarian National Foundation for Scientific Research under grants OTKA T046993 and T043657. This work was also supported in part by a guest-professorship at the University of Paderborn.