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

SOLUTION OF PELL EQUATIONS BY THE HELP OF LINEAR RECURRENCES BY PÉTER KISS (Summary) Let A, B, G

0 and Gj be fixed integers such that AB ^ 0 and G

0, G

x are not both zero. The infinite sequence G

0, G

x, G

2, ... of integers, for wich G

n = A -Gn.j — B -G

n_

2 for n > 1, is called second order linear recurrence and we shall denote it by G = G(A, B, G

0, G

x). If D = A

2-4B ^ 0, then the terms of G can be written in the form _ ba

n —c/?

n n — a ' a— p where a and are the roots of the polynomial f(x) = x

2 — Ax 4- B and (i) b = Gj — G

0/3 and c = G, — G

0a. The sequence H, defined by H

n = ba

n + c/?

n, is called the associate sequence of G. The sequence H is also a linear recurrence having parameters A, B, H

0 = 2G

X —AG

0 and H\ = AG

X —BG

0. Some connections are known between the Pell equations of the form x

2 —Dy

2 = N and second order linear recurrences. For example V. E. Hoggatt [4] proved that the all integer solutions of the eqqation x

2 — 5y

2 = = ±4 are the numbers (x;y) = (±L

n;±F

n), n = 0, 1,2, ..., where L(l, —1, 2, 1) and F(l, —1, 0, 1) is the well knowm Lucas and Fibonacci sequence, respectively. Some similar results were obtained by I. Adler [1, 2] E. M. Cohn [3] and V. Thébault [7] for the equation x

2- 2y

2 = ±1. The purpose of this paper is to look for solutions of some classes of Pell equations. We prove the following theorems. THEOREM 1. Let N (^ 0) and a (> 0) be integers. If the equation x

2-(a

2+l)y

2 = N 823