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

72 Tamás Herendi and Attila Pethő min( I

Xi —x, I) Let l = ——2 where x<\ ... x^-ij are the roots of the polynomial F

n-i(x). We have / > 0 by Lemma 2. If min — > l then Fn-1 M n i=1 a r ~ > /M from where it follows that -^-rfr > 6

n and there exist only finitely many z L 2 J 6 E Z which is suitable for this. This together with the fact, that p- is bounded implies that there exist only finitely many possible a, 6 pair. Let min ([x^ — < /. Obviously among the \x{ - < l inequalities hold i only one. Let suppose that |x

í

0 — < /. Then > íh a X*0 - p hence using (11) we get B-M /[V]. b > n — k a X

l° ~ 62 As 6- /

n_ 1 (a, 6

2) ^ 0 so ^ p-. We assumed n —fc > 4, hence the theorem of Roth on approximation of algebraic numbers [3] implies that there exist only finitely many suitable a, 6 pair for this approximation if ÍQ is given. The number of the roots of F

n-i(x) are finite so there exist only finitely many possible a, 6 pair and so there exist only finitely many possible A values, (b) Let gcd(n, k, 12) = m > 1. Then by Lemma 7 gcd (

y n — 1 mod 2 •/

n_

x (x,y

2) ,y k — 1 mod 2 A-i (z>!/

2)) = îTlmod

2-/

m-1 (x,t/

2). Hence there exist #1(2:, 2/), C^? v) £ such that yn-1 modí.^ (arY) =

í/

1(x,y).y

m1 mod

2-/

m-! (x,</

2), mod2 (x,y

2) 1 mod

2 (z,y

2). We have by Lemma 4 5 -y"

11 mod

2'f

m-i (x,y

2) =gi(x,y)-y

m~

l mod

2 • f

m-1 (x,y

2) .