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 TAMÁS HERENDI and ATTILA PETHŐ* Abstract. The reducibility of the trinomials in the form X

N — Bx

k — A are examined. It is shown, that among the trinomials in the same class (i.e. somé of the Parameters A,B,k and n are fixed) there are only finitely many wich has quadratic factor. 1. Introduction Let us consider the trinomial x

n — Bx

k — A. Ribenboim [4] has shown that if k = 1 then for a fixed n and B there exist only finitely many A for which the trinomial is divisible by a quadratic polynomial and similarly if n and A is fixed then there exist only finitely many B for which the trinomial has a quadratic factor. He used in the proof elementary steps only. Schinzel in [5] then presented a much more generál result in which he proved among ot her s that for fixed A there exist only finitely many n,k,B for which the trinomial is divisible by any polynomial. He could prove similar result for fixed B too. His proof is however not an elementary one. We are also able to generalize Ribenboim's result extending his proof but keeping its elementaryness. Our result is less generál than Schinzel's result. We prove the following theorems: Theorem 1. Let be given k G N and A 6 Z \ {0}, then (a) there exist only finitely many, effectively determinable polynomials in the form x

n - Bx

k - A, where n G N, B G Z \ {0} and gcd(fc, n, 12) = 1, for which x

2 - bx - a I x

n - Bx

k - A with a, è G Z. (b) if gcd(k, n, 12) > 2 where n G N then there exist only finitely many effectively determinable polynomials in the form x

n — Bx

k — A , where B G Z \ {0} for which x

2 — bx — a\x

n - Bx

k — A for an a, b G Z pair. * Research (partially) supported by Hungárián National Foundation for Scientific Research, grant No. 1641.