Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1984. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 17)

II. TANULMÁNYOK A TERMÉSZETTUDOMÁNYOK KÖRÉBŐL - Kiss Péter: Feli egyenletek megoldása lineáris rekurzív sorozatok segítségével

has integer (x;y) solutions, then all solutions are obtained by (x;y) = = (± (Gf

2

n + aG

2n+

1) ;±G

2n+

1), n = 0, 1, 2, ..., using finitely many linear recurrences G(2a, —1, G

0, G^. These sequences satisfy the conditions 0 G

1 < 2a j/N or 0 s Gj < (2a

2

+1) f- N/(a

2+ l) according as N > 0 or N < 0. THEOREM 2. Let N 0) and a ( > 2) be integers. If the equation x

2 — (a

2 — 4)y

2 = 4N has integer (x ;y) solutions, then all solutions are obtained by (x;y) = (±H

n; + G

n), n = 0, 1, 2, ..., using finitely many linear recurrences G(a, 1, G

0, GjJ and their associate sequences H. These sequences satisfy the conditions 0 ^ Gj, < |/N or 0 g G

x < a / —N/(a

2—

4) according as N > 0 or N < 0. THEOREM 3. Let N (^ 0) and a (> 0) be integers. If the equation x

2 — (a

2 + 4)y

2 = 4N has integer (x;y) solutions, then all solutions are obtained by (x;y) = = (±H

2

n; + G

2

n), n = 0, 1, 2, ..., using finitely many linear recurrences G(a, — 1, G

0, Gj) and their associate sequences H. These sequences satisfy the conditions 0 ^ G

0 < a /N or 0 ^ G

0 < (a

2+ 2) / — N/(a

2 + 4) according as N > 0 or N < 0. THEOREM 4. If the pairs (x;y) = ( + H

2

n; + G

2

n), n = 0, 1, 2, . . ., are solutions of the equation x

2—(a

2 + 4)y

2 = 4N, then the numbers (x;y) = = (±H

2n+

1; + G

2n+

1), n = 0, 1, 2, ..., are solutions of the equation x

2— (a

2 + 4)y

2 = — 4N. The proofs of the theorems are based on the following two results

: 1. If (A, —1, G

0, G

x) is a linear recurrence and k = G

0

2 -f AGoGj

2 — Gj

2, then G„

2 + A.G

n.G

n+

1-G

2

+

1 = (-l)".k for every integer n ^ 0. 2. Let G = G(A, B, G

0, G,) be a linear recurrence and let D = A

2 — 4B ^ 0. If H is the associate sequence of G, then H

n

2 —D .G

n

2 = 4bc-B

n for every integer n ^ 0, where b and c are defined in (i). 824