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

ZAY , B., An application of the continued fractions for ... in solving some types of Pell's equations

An application of the continued fractions for in solving some types of Pell's equations BÉLA ZAY Abstract. In this paper we study the positive solutions of the Diophantine equation x

2 — Dy

2 — N, where D and \N\ are natural numbers, |7V|<\/D and D is not the square of a natural number. Let \ZD=-(a

0,ai,...,a,) be the representation of ^/D as a simple continued fraction expansion. We prove that if the n-th convergent to \/D is |

J

L=(a

0

r..,a

n), then ^O + 2)3 + r=2//

s _].//(„ + i)

3 + r + (-1)

5 +

1 H

n, + r and #(n+2)j + r=2//

3_

1/i(

n+

1)

J + r+(-l)

J+

1K'

t

V3 + r . In cases of D-(2k+l)

2 -4 (for any k>2), D-(2k)

2-4 (for any fc>

3), D-k

2 1 (for any k>

2) and D=fc

2+

1 (for any A:>l) we give all positive solutions of x

2 ~Dy

2 = N (|Af|<\/D) with the help of Binet formulae of the sequences (H

n, + r) and (/i„, + r) (for any r=l,2,...,s). Introduction In this paper we consider the equation (1) x

2 - Dy

2 — N and its solutions in natural numbers, provided D and N are rational integers, D > 0, furthermore D is not the square of a natural number. Many authors studied these Diophantine equations. Among others D. E. FERGUSON [1] solved the equations x

2-by

2 = ±4, V. E. HOGATT, JR. and M. BLCKNELLJOHNSON [2] solved the equations (2) x

2 - (A

2 ± 4)y

2 = ±4 where A is a fixed natural number. K. LIPTAI [4] proved that if there is a solution to (1) then all solutions can be given with the help of finitely many, well determined second order linear recurrences. Research supported by Foundation for Hungarian Higher Education and Research and Hungarian OTKA Foundation Grant No. T 16975 and 020295.