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 5 These with (8) imply that Sj+5 ~ + 2 + Sj2 — Sj7 = ^j-l — = ~~ ^i-3 + — — Sj+l * Sj3 + ^ - Sj—4

, which proves (7) with n = j — 7. By (8), we have <5*7 = <$2-3 + 1 = 25

5 — Si, S 9 — 5"2-4+l = £7 + — -^2 = Sq + 25

S — S 2 — 5i and «5*11 = 52.5 + 1 = 5g + 5*7 — S3 = S6 + 4,5*5 ~~ S3 — S2 ~ 25i. Finally, by using (6) and the facts 8

2 + l

2 = 7

2 + 4

2, 10

2 + 5

2 = ll

2 -f 2

2 and 12

2 -f l

2 = 9

2 + 8

2

, we have S$ = S7 S4 — Si = 2S$ + 5*4 — 25i

, Sio = Sn + S 2 — S 5 = iSg + 35

5 — 5*3 — 261 and S12 = S

9 + Ss - Si = Se + 4S

5 + S

4 - S

2 - 45i

. Thus, to complete the proof of the lemma , by using (1), (4), (5) and (7), it is enough to prove that either Si = S2 — S3 = S^ — = S& = 0 or (9) Sj = j

2 + 1 for 7 = 1,2,3,4,5,6. Repeated use of (1), using the multiplicativity of /, gives Si = /(l

2 + 1) = /(2), (10) <>2 = /(2

2 + 1) = /(5) = /(

l

2 + l

2 + 3) = /(2) + /(3), (11) 5

3 = /(

3

2 + 1) = /(10) = /(2)/(5) = /(2)

2 + /(2)/(3). and thus /(11) = /(

2

2 + 2

2 + 3) = /(5) + /(6) = /(2) + /(3) + /(2)/(3).