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

On a conjecture about the equation jynx ßrny _ j^vrxz ALEKSANDER GRYTCZUK Abstract. Let A be a given integral 2x2 matrix. We prove that the equation (•) A

m

i +A

m

y = A

m

z has a solution in positive integers x,y.z and m>

2 if and only if the matrix A is a nilpotent matrix or the matrix A has an eigenvalue i + _ 1. Introduction First we note that (*) is equivalent to the following Fermat's equation (1) X

M + Y

m = Z

M, m > 2, where X = A

x

, Y - A

y and Z = A

z

. It has been recently proved by A. WLLES [12], R. TAYLOR and A. WILES [11] that (1) has no solution in nonzero integers X,Y, Z if m > 2. But, in contrast to the classical case, the Fermat's equation (1) has infinitely many solutions in 2 x 2 integral matrices X, Y, Z for m — 4. This fact was discovered by R. Z. DOMIATY [2] in 1966. Namely, he proved that, if !)• y=(° o) o where a, b

, c are integer solutions of the Pythagorean equation a

2 -f b

2 — c

2, then x

4 + y

4 - z

4. Other results connected with Fermat's equation in the set of matrices are given in monograph [10] by P. RlBENBOIM. In these investigations it is an important problem to give a necessary and sufficient condition for the solvability of (1) in the set of matrices. Such type results were proved recently by A. KHAZANOV [7], when the matrices A, Y, Z belong to SL

2{Z), SLz(Z) or GL$(Z). In particular, he proved that there axe solutions of (1) in X, Y.Ze SL

2(Z) if and only if m is not a multiple of 3 or 4. We proved