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

62 Tamás Herendi and Attila Pethő Theorem 2. Let be given n,k £ N and B £ Z \ {0} then (a) if gcd(n, k, 12) = 1 and n — k>4 then there exist only finitely many A £ Z for which x

2 — bx — a \ x

n — Bx

k — A for an a, b £ Z pair; (b) if gcd(n, k, 12) > ß and n — k > 4 then there exist infinitely many A G Z for which x

2 - bx — a | x

n - Bx

k — A for an a, b £ Z pair, but except for finitely many values ail the possible values of A is explicitely expressable as a sériés. Remark. Using properties of curves of genus at least 1, we were able to handle the case n — k < 4 too. As Schinzel's resuit are more generál and our proof is not elementary, we omit the détails. 2. Auxiliary resuit s Let the polynomial sequence defined as follows: ^(z) — F\(x) = 1, and if n > 2 then F

n(x) — F

n_i(x) + x • F

n_

2(x). Let define the polynomial sequence 3

5 fn(

x,y) — MÍ)Remark. From Lemma 2 you can see that f

n(x, y) is really a polynomial and not a rational function. Lemma 1. The sériés {i

7

,

n(x)}^L

0 has for any 1 < k < n the following properties: (a) F

n(x) • F

k-i(x) = F

ni(x) • - (-1)* • x

k • F

Bf

c_i(®); (b) F

n(s) = F

nk(x) • F

k(x) + x • F

n_

k^i(x) • F

k-i(x). PROOF. We prove only property (a), because the proof of (6) is similar. Let k — 1. Then n > 2. The equality in this case is true because F

n(x) • F

0(x) = F

n^(x) • F^x) + a; • F

n„

2(x), where io(a:) = Fi(x) = 1, and this is exactly the defining équation of F

n if n > 2. Let now k > 2 and suppose that for every 0 < i < k the equality holds. We know that (I) F^x) • F

k(x) = F

n(x) • (F

k^(x) + x • F

k_

2(x)) (II) Fn^fx) • F

k(x) = F

n_i (x) • (Ffc-if®) + a: • F

k.

2(x)) and (-1)* • z*"

1 - F

n.

k+

1(x) = (-l)

k • x

k~

x • (F

n_

k(x) + x • F

n.

k^(x))

, which is equal to (III) (

~l)

k • x

k . F

n.

k.

x (x) = (-l)

k • x

k~

l • (F

n_

k+

l (x) + x • F

n.

k(x))

.