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
70 Tamás Herendi and Attila Pethő \ nb 2 -f v D ) 7ÏÏ n+1 mod 2 •b n+1 Hence A > ..n — k — 1 mod 2 fn-k-1 (a,& 2) >T bk-l mod 2 . f k_ x fon-k-2 = b n—2k—2 Because of the monotonity of the exponential function and the finiteness of the number of the possible a-s there exists an upper bound for n depending only on A. Let examine now the case b 2 < max (4a 2,m a). There exist only finitely many b satisfying the inequality. For these values we apply Theorem 3.1 from [2] and get (10) f n(a,b 2) > 1^1 n — 1 — cx log(n — 1} where c\ is efFectively computable and depends finally on a and b. As we have only finitely many possibilités for a and b, c\ is a constant and for n large enough the exponent in (10) is positive. By aresult of Dobrowolski (see in [1]) \Ui \ > C2 > 1 holds for any quadratic algebraic integers which are not roots of unity, hence by (10) similarly to the previous case n is bounded. So there exist only finitely many possible ns satisfying the assumptions in the theorem from which using Lemma 5 follows the statment of the theorem. Suppose now that gcd(&, n) > 1, but gcd(&, n, 12) = 1. If a and b satisfy the assumptions in the theorem then b k~ l mod 2 • fk-i (a, ö 2) isn't zéro otherwise by Lemma 4 (a) b n~ l mod 2 • / n_i (a, b 2) would be zéro, which is impossible. Hence bn-k-1 mod 2. f n_ k i A = a k(-iy bk-1 >d 2 fk-i (a,b 2) and the proof of the theorem in this case is the same as in the previous case. (b) In this case we can divide the possible a,b pairs into two sets. In the first set b k~ l mod 2 • fk-1 (a, b 2) ^ 0. Similarly to the previous two cases there exist only finitely many solution for B. In the second set b k~ 1 mod 2 • / f c_ 1 (a,b 2) = 0. Then by Lemma 4 (a) b n~ l mod 2 • / n_ x (a,b 2) = 0 . This is possible if and only if one of the following statements holds: 1. n is even and b = 0, or
