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

12 Béla Zay By the formulas (9), (10), (13) and (14) we can verify that 3n+l _ /0371+1 jj - 3n+l , /0372 + 1 v _ « P " 6ti + 1 — a + P , Aßn + 1 — 7 a — ß (et - 1 )a

3n+

1 + (/3 - l)/3

3n+

1 #6n + 2 — A 6n+2 — 2 (a - l)a

3n+

1 - (ß - l)ß

3

n+

l 2

(a - /5) Q3n+2 _ ^Sn + 2 rr _ 3n+ 2 , /j3n + 3 r- _ J26n + 3 — a + ß , A

6

n+3 H !6n + 4 — a — ß (a - 2)a

3n+

2 + (/3 - 2)ß

3n+

2 A 6n+

4 — ( 2 a - 2)o

3n+

2 - (/3 - 2)/?

3

n+

2 2(a-/5) for n > 0 and Q3n+3 ^3n + 3 c*

3n+

3 - /5

3

n + 3 Hßn+b ~ 5 Äßn+5 H 6n + 6 = Äßn + 6 = 2 ' 2 (2q - l)a

3

n+

3 + (2/3 - l)/?

3

n+

3 2 ' (2q - l)g

3n+

3 - (2ß - l)/3

3

n+

3 2(0"—^) ' for n > — 1. From these equations we obtain, that r 1, for r = 5 ) rr

2 n r'2 _ 4, for T = 1 or 3 ^6n+r - ^Afln + r - x _ 2

A. for r = 2 C>+1 for any n > 0. Prom (25) and (16) we can easily verify that the statements of Theorem 1. are valid. The proofs of the Theorem 2., Theorem 3. and Theorem 4. are carried out analogously to the proof of the preceding theorem. For brevity we write only few formulas (without details) in this proofs. Proof of Theorem 2. y/D = yj{2k)

2 - 4 = (2k - 1,1, k - 2,1, k - 2), for jfc > 3 rv2n + l _ /32n+l rr _ 2n +

1 , ,q2n+l r- _ " P fl 4n + l — ° + P 1 ^4n+l — , a — ß