Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

GRYTCZUK, A., On a conjecture about the equation Amx + Amy=Amt

62 Aleksander Grytczuk in [4] a necessary condition for the solvability of (1) in 2 X 2 integral matrices X".,Y,Z having a determinant form. More precisely, we proved (see [4], Thm. 2) that the equation (*) does not hold in positive integers x

, y,z and M. H. LE and CH. LI [8] proved the following generalization of our a + d > 0 and det A — ad — be < 0, then (•) does not hold. In their paper they posed the following Conjecture. Let A be an integral 2x2 matrix. The equation (*) has a solution in natural numbers x

, y,z and m > 2 if and only if the matrix A is a nilpotent matrix. A corrected version of this Conjecture was proved by the same authors in [9]. In the present paper we prove the following Theorem. The equation (•) has a solution in positive integers x,y,z and rn > 2 if and only if the matrix A is a nilpotent matrix or the matrix A has an eigenvalue a = . We note that the condition matrix A has an eigenvalue a = ^ is equivalent to Tr A = det A = 1 (cf. [9]). On the other hand it is easy to see that the condition det A = 1 implies that the matrix A cannot be a nilpotent matrix, thus the original Conjecture of M. H. LE and CH. Li is not true. We also note that X. CHEN [1] proved that if A

n is the companion matrix for the polynomial f(x) = x

n - x

n~

1 — ... — x — 1 then the equation (•) with A — A

n has no solution in positive integers x,y,z and m >2 for any fixed integer n > 2. Futher result of this type is contained by [5]. Namely, we proved the following: Let A = ( fl(j)

nX

n be a matrix with at least one real eigenvalue ot > y/2. If the equation . Another proof of this cited result was given by D. Frejman [3]. be a given integral matrix such that r = Tr A = (2) A

r + A

s = A has a solution in positive integers r,s and t then max{r — t,s — tj = —1. From this cited result one can obtain the corresponding results of the papers [1], [3], [4], [8] as particular cases.