Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

HERENDI, T. and PETHŐ, A., Trinomials, which are divisible by quadratic polynomials

66 Tamás Herendi and Attila Pethő Using this we pro ve that ifl<i<n — 2, then (3) c

n_

2_; - b

{ mod 2 • b

2) - Bb

k~

n+

i ™

d 2 • /

f

c_

n_

t(a, b)\ By (2) it is easy to see that (3) holds for i — 1, 2. Let 2 < i < n — 2 and suppose that (3) holds for every j with 1 < j < i. Then by (2) we get Cn-2-i = « • C

n-2-(i-2) + b • C

n_2_(i_l) - <$n-t,fc ' B = a • Cl + b • C

2 - 6

n-i

tk • B _ bi-2 mod 2 . C

s _ B . bk-n+i-2 mod 2 . ^ + £ . ^ where Ci = mod2 . f._

2(

ay) - B . B*-"*2 mod

2-A_

n+

i_

2(a,6

2) C

2 = 6*"

1 mod

2 • /i_i(a,b

2) - B • B

k~

n+

i~

x mod

2 • /,_

n+

í_!(a, 6

2) C3 = 62(i-i mod2) . + f

l . f

k_

n+

l_

2(a

:b

2) C

4 = fr

2«-"*'1 mod

2) • A_

n+

l_i(a,6

2) + a • /

f

c_

n+

I_

2(a, ö

2). From this by Lemma 3 we get (3). Using (2) and (3) we obtain 0 = a • ci + b • c

0 - öl,* • B = a • (b

n~

3 mod

2 • /

n_

3(a, 6

2) • 6

fc_3 mod

2 • A_

3(a,6

2)) + +6 • a • (62mod 2 • /

n_

2 (a, 6

2) • ^2mod 2 • /

f

c_

2 (a, 6

2)) - 6

lt

k • B = hn-3 mod 2 . ^2(n-2 mod 2) (

a, 6

2) + a •/

n_

3 (a, 6

2 )) . ^-3 mod 2 . ^2(k-2mod 2) ^ &2) + f

l . ^ ^ &2)) + Using Lemma 3 we get (4) 0 - b

n~

l mod

2 • fn—i (a, b

2) — B • b

k~

l mod

2 • (a,6

2) , which proves the first assertion. This imphes b

k~

l mod 2 • 1 ( a, b

2) = 0 if and only if b

n~

l mod 2 • /

n_i (a, b

2) = 0. By (2) and (3) (5) A = a • (b

n~

2 mod

2 • /

n2(a,6

2) - B • b

k~

2 mod

2 • A_

2(a,6

2)) . (b) If b

k~

l mod 2 • f

k-1 (a, 6

2) ^ 0 then from (4) we get bn-1 mod 2.^ (

aj

62) (6) 5 = mod 2 . f

k_

l (a, 6

2)