Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

CHUNG, P. v., Multiplicative functions satisfying the equation f(m2+n2) = (f(m))2 + (f(n))2

Multiplicative functions satisfying the equation /(™

2 + n

2) = (/(m))

2 + (/(n))

2 PHAM VAN CHUNG Abstract. Purpose of the present paper is to characterize multiplicative functions / which satisfy the equation f(m

2+n

2)=(f(m))

2+(f(n))

2 for all positive integers 171 and n. 1. Introduction A "multiplicative function" is a function / defined on the set of the positive integers such that f(mn

) = f(m)f(n) whenever the greatest common divisor of m and n is 1. The function / is called "completely" multiplicative if the condition f(mn) = f(m)f(n) holds for all m and n. Claudia A. Spiro [2] proved that if a multiplicative function / satisfies the condition f(p + q

) = f(p

) + f{q) for all primes p, q and f(po) ^ 0 for at least one prime po

, then f(n) = n for each positive integer n. Replacing the set of primes by the set of squares, in [1], we have investigated the multiplicative functions / satisfying the condition (A) /(m

2 + n

2) = /(m

2) + /(n

2) for all positive integers m, n. We have shown that if / ^ 0 is mult ip he at ive, then / fulfills the condition (A) if and only if either (A-l) /(2

f

c) = 2

k for all integers k > 0, (A-2) f(p

k) = P

k f°

r primes p = 1 (mod 4) and all integers k > 1, and (A-3) f(q

2

k) = q

2

k for all primes q= 3 (mod 4) and all integers k > 1 or (a-l) /(2) = 2 and /(2

k) = 0 for all integers k > 2, (a-2) f(p

k) = 1 for all primes p = 1 (mod 4) and all integers k > 1, and