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 BUI MINH PHONG* Abstract. We prove that if a multiplicative function / satisfies the equation /(n

2 + 7Ti

2+3) = /(n

2+l) + /(m

2+2) for all positive integers n and m, then either /(n) is the identity function or /(n

2+m

2+3) = /(n

2+l)=/(m

2+2)=0 for all positive integers. Throughout this paper N denotes the set of positive integers and let A4 be the set of complex valued multiplicative functions / such that /(1) = 1. In 1992, C. Spiro [3] showed that if / E A4 is a function such that f(p + q) = f{p) + f{q) f°

r ^ primes p and q, then f(n) = n for all n E N. Recently, in the paper [2] written jointly with J. M. de Köninck and I. Kátai we proved that if f E A4, f(p+n

2) = f(p) + f(n

2

) holds for all primes p and n E N, then f(n) is the identity function. It follows from results of [1] that a completely multiplicative function / satisfies the equation f(n

2 + m

2) = /(n

2) + f(m

2) for all n, m E N if and only if /(2) = 2, f(p) = p for all primes p = 1 (mod 4) and f(q) = q or f(q) - —q for all primes p = 3 (mod 4). The purpose of this note is to prove the following Theorem. Assume that f E A4 satisfies the condition (1) / (

n

2 + m

2 + 3) = f{n

2 + 1) + /(m

2 + 2) for all n,m E N. Then either (2) f(n

2 + 1) = f(m

2 + 2) = f{n

2 + m

2 + 3) = 0 for all n,m£ N, or f(n) = n for all n E N. Corollary. If f E M satisfies the condition (1) and /(n§ + 1) ^ 0 for some tiq E N, then /(n) is the identity function. First we prove the following lemma. Lemma. Assume that the conditions of Theorem 1 are satisfied. Then either (2) is satisfied for all n E N or the conditions /(n

2 + 1) = n

2 + 1, f(m

2 + 2) = m

2 + 2 and ^ /(n

2 + m

2 + 3) = n

2 + m

2 + 3 * Research (partially) supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 2153 and T 020295.