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

8 Bui Minh Phong Then we have <7~1 / / 2 o\ \ E I H^KIK-m) (m

2+3,

9) = X 1 1 ' / —m

2 — 3\ \ 1 ( /-3 „ 2 V V <i )) 2 V V ? m =0 \ x * ' ' ^ \ ^ 9|m2+3 1 + = ^ ( ^ - ( - 2 - 2 ^ 3 By our assumption, the last relation implies that jj*4

9(3) > 1. Thus, there are integers m G {1,..., q — 1}, 1 < n\ < q

a — 1, (ni ,q) = l and 1 < n

2 '•= q

a — ni < q

a — 1 such that n

2 + m

2 +3 = q

aNi (i = 1,2). It follows from the above relations that q

a(N

2 - Ni) = (q

a - m)

2 -n

2= q

2

a - 2q

an

u that is N

2 - Ni = q

a - 2ni

. Since (ni

, q) = 1, we obtain that at least one of N\ or N

2 is coprime to q. Let n G {ni,n

2} and N G {Ni,N

2} such that n

2 + m

2 + 3 = g

aiV, (N,q) = 1. Then a > 2 imphes that N < — - q<* < q

a. (^-l)

2 + (?-l)

2+3 <r 'Thus, A7(<Z

a) = f(N)f(q

a) = /(iv

g

a) = f(n

2 + rn

2 + 3) = n

2 + m

2 + 3 = iVg

a, which shows that f(q

a) = q

a as we wanted to establish. To complete the proof of the theorem, it remains to consider the cases q = 2 and q = 3. Let q— 2 and T = 2

a

, where a > 3. Since —7=1 (mod 8), we have —7 is a quadratic residue modulo 2

a and therefore there exists n

a G [0, 2

Qf_

1 — 1] such that n\

+7 = n\+2

2 +3 = 0 (mod 2

a), and consequently, [n

a + 2

a_

1]

2 + 7 EE 0 (mod 2

a). Define JVi,