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

4 Bui Minh Phong simultaneously hold for all n,m 6 N. Proof. From (1), we have f{n

2 + 1) + /(m

2 + 2) = /(m

2 + 1) + f(n

2 + 2) for all n, m E N, and, so (4) f(n

2 + 2) - f(n

2 + 1) = /(3) - /(2): = D for all n G N. Thus, the last relation together with (1) implies that (5) * f(n

2 + m

2 + 3) - f{n

2 + 1) + /(m

2 + 1) + holds for all n,m £ N. Let Sj : = f(j

2 + 1). It follows from (5) that if k,l,u and Í;£

N satisfy the condition k

2 + /

2 = u

2 + v

2, then f{k

2 + 1) + /(/

2 + 1) + D = f{u

2 + 1) + f(v

2 + 1) + D, which shows that (6) k

2 + I

2 = u

2 + v

2 implies Sk + Si — S

u + S

v. We shall prove that (7) S

n.

(-12 = Sn+

9 + Sn+

8 + ^N+7 — ^N+5 — ^N+4 ~ SN+3 + ^N holds for .all n £ N. Since (2j + l)

2 + (j - 2)

2 = (2j - l)

2 + (j + 2)

2 and (2j + l)

2 + (i - 7)

2 = (2i - 5)

2 + (i + 5)

2, we get from (6) that (8) + Sj-2 = 5

2

j-1 + Sj+2 and ^j+l + Sj7 = + ^j+5-