Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
PHONG, B. M., A characterization of the identity function
A characterization of the identity function 9 and N 2 by n 2 a + 7 = 2 0 tN l and [n Q + 2 a~ 1] 2 + 7 = 2 aN 2. We easily deduce from these two equation and the fact 7 < T = 2 a that N x < 2 a, N 2 < 2 a and N 2 - N t = n a + 2 a~ 2 . It follows from the last relation and the fact 2 does not divide n a that one of Ni or N 2 is odd, and so f(2 a) = 2 a . Finally, let q = 3 and T = 3 a , where a > 1. We consider the congruence n 2 + 2 = 0 (mod 3 a). Similarly as above, one can deduce that there are positive integers n, N E N such that n 2 + 2 = 3 aN , (N, 3) = 1 and N < 3 a . Thus these together with (18) implies that /(3") = 3 a. The theorem is proved. References [1] P. V. CHUNG, Multiplicative functions satisfying the equation /(m 2 + n 2) = f(m 2) + /(n 2), Math. Slovaca, 46 (1996), No. 2-3, 165-171. [2] J.-M. DE KÖNINCK, I. KÁTAI and B. M. PHONG, A new charateristic of the identity function, J. Number Theory (to appear). [3] C. SPIRO, Additive uniqueness set for arithmetic functions, J. Number Theory 42 (1992), 232-246. Bui MINH PHONG DEPARTMENT OF COMPUTER ALGEBRA EÖTVÖS LORÁND UNIVERSITY MÚZEUM KRT. 6-8 H-1088 BUDAPEST, HUNGARY
