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 A

M

I -f A

M

Y-A

M

Z 69 Substituting r = 0, a

2 + be = d

2 + be = 0 to (30) we obtain that A

2 = 0, that is the matrix A is a nilpotent matrix with nilpotency index two. Now, we suppose that the matrix A is nilpotent matrix, i.e. A

k = 0 for some natural number k > 2. Then it is easy to see that (•) is satisfied for all positive integers x,y, z,m > 2 such that mx > k

, my > k, mz > k. Suppose that the matrix A has an eigenvalue a = . Then it is easy to check that a

2 — = £ is a third root of unity. By an easy calculation we obtain if n = 6k, c : if n = 6k + 1, if n = 6k + 2, 1' if n = 6k + 3, if n = 6k + 4, if n - 6k + 5. Applying (31) we obtain that (*) is satisfied if and only if the following relations are satisfied (32) mx = ri( mod 6), my = r

2( mod 6), mz — r

3( mod 6), where (ri,r

2,r

3> = {0, 2, l), (0,4, 5), (l, 3,2), (l, 5, O), (2,4, 3), (2, 0, l), (3,1,2), (3, 5,4), (4, 0,5), (4, 2, 3), (5, 0,1), (5. 3,4). The proof of Theorem is complete. From the proof of Theorem we get the following Corollary. All soluitions of the equation(*) in natural numbers x,y,x and m > 2, when the matrix A has an eigennvalue a = l + are given by the congruence formulas (32) with the above restrictions on (ri,^,^) and if the matrix A is a nilpotent matrix with nilpotency index k > 2 then (*) is satisfied by all positive integers x,y,z,m > 2 such that mx > k.my > k and mz > k. Remark. We note that Theorem with Corollary is equivalent to the result presented by M. II. LE and CH. LI in [9], but our proof is given in another way and it gives more information about the impossibility of the solvability of (•) in the cases mentioned in Lemma 3, 4, 5.