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 7 Assume now that /(2) = 2. In this case we have /(5) = 2 + /(3), /(8) = 2 + 2/(3). We shall prove that /(3) = 3. It follows írom (15) and using the fact /(37) = /(

l

2 + 6

2 + 3) - /(3) = /(5)/(8) - /(3) = 2/(3)

2 + 5/(3) + 4 that (17) 2/(3)

2 - 4/(3) - 6 = 0. On the other hand, from (4) we infer that /(6)/(ll) - /(3)/(13) = /(66) - /(65) = /(3) - /(2), consequently 3/(3)

2 - 7/(3) -6 = 0. This together with (17) proves that /(3) = 3, and so (10)—(17) imply that Sj=j

2 + 1 (J = 1,2,3,4,5,6). This completes the proof of (9) and so the lemma is proved. Proof of the theorem In the proof of the theorem, using the lemma, we can assume that (3) is satisfied, that is f(n

2 + 1) = n

2 + 1, /(m

2 + 2) = m

2 + 2 and ^

1

8^ f(n

2 + m

2 + 3) = n

2 + m

2 + 3. It is clear from (18) that f(n) = n for all n < 7. Assume that f(n) = n for all n < T, where T > 7. We shall prove that f(T) = T. It is clear that T must be a prime power, that is T = q

a with a G N and some prime q . It is easily seen that if a = 1, then q > 7 and there are positive integers ra, m < ^ such that n

2 + m

2 + 3 = qN

, (5, JV) = 1 and N < q. Thus, we have f(q) = q. Assume now that a > 2 and q > 3. We consider the congruence n

2 + m

2

+3 = 0 (mod q

a). Let 2 ^,(3):=|l<m<qr-l: = l| .