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

Trinomials, which are divisible by quadratic polynomials 69 f

k-i (x,y

2) then y | y

m~

l mod

2 • /

m_j (x,y

2) and vice versa. The leading coeffitient of /

m_i (x, y

2) is différent &om 1 if and only if m is even and in this case it is equal to m. But then the leading coeffitient of f

n_i (x, y

2) is equal to n and the leading coeffitient of i (x,y

2) is equal to k. Lemma 8. Let D = y • (4x + y), U

x = and U

2 = . Then n+1 mod 2 where / \ Tl-tL moa i, I rtji I / [y+y/D-U^ -U\ 2 j- (y-yfD-U. •c, n+1 mod 2 [n±i] / / \ n+1 mod 2 + • U*) c = y/D PROOF. It is easy to see that f

n{x,y) has the property F

n+

2(x,y) = (2x + y) • fn(

x, y) — x

2 • fn-2{

x, y)- From this similarly to the method used in Lemma 2 for F

n(x) we get the statement of the Lemma. 4. Proof of the theorems Proof of Theorem 1. (a) Let suppose that gcd(fc,n) = 1. At first we show that there exists an effectively computable upper bound for the possible n values. If A / 0 is given then by Lemma 2 (b) using the définition of f

n b

k~

l mod 2 • 1 (a, b

2) /0. Then by Lemma 4 (b) un-k-1 mod 2 f (

n u2\ a M and a^-l)*

* , d

2 f 7* ' [ A. Let assume now that n is given and a is fixed and suppose that b

2 > 4a

2. Then if we Substitute x by a and y by b

2 in Lemma 8, we obtain D > 0, Ui > a

2 and U

2 < 1. Then there exists M

a constant, such that |/fc-i (a,6

2)| < IM

ab

k~

l I if b / 0 and from Lemma 8 follows that there exists m

a, n

a and c

a > 0 such that if n > n

a and \b\ > m

a thén as U\ > ~ fn («,»*) = g) n+1 mod 2 • c n+1 mod 2 flL±lj \ n-t-i moa i b

2 + y/D -Ui) • U{ —