Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

H.-MOLNÁR, S., Approximation by quotients of terms of second order linear recursive sequences of integers

24 S. H.-Molnár 3

r

í Example, t' is a non-real quadratic algebraic integer. -3+tV31 Let t' = ai, where cvi is the root of x + Zx + 10 = 0,i. e.|c*i| = yiÖ. Since |ai| = yTÖ = 6+/36+

* _ 3 |

a

i| ^ where Q is a root 0f x

2 — 6a,* — 1 = 0 and |ai[ = a — 3. Calculating with the sequences G(6, 1, —3, 1) . This ^±1 Hr, < 2VIOIÍ; and H(6,

1, 0, 1), A,'o = 2 and c

0 = \/40 and thus approximation is the best. A

t

h Example, a is a complex, non-algebraic quadratic integer. Ax

2 + 5x -f 6 = 0, |cvi| = -5-V25-96 |

0

l| = ^24 _ 1

4+743+4. 4 — 0 0 1 = — 1, where a is root of the equation x

2 - Ax - 2 = 0. A = A, B = 2, G'(4, 2, -2, 2) and #(4,2,0,4), = 2 - = 1, 535669821 ..., c

0 = 0, 5573569115 .... Calculating with the sequences G*(4, 2, —1,1) and H*(4, 2, 0, 2), = = 2

f

c°-c

0 = 1,615905915.. . . Proof of Theorem. By (1) we can write G

n+i — a\a

n+

1 — b\ß

n+

l and H

n = aa

n — bß

n for any n > 0, where G

1 — Goß psB — qrß psB — qra ai = ^— = „—

bl = a-ß a -ß ' qsB - 0ß qsB a = b = a-ß ' qsB a — ß a — /?' a — ß Suppose that for an integer n > 0 and the positive real numbers c and k we have (2) t G 71+1 Hr < 1 c\H

r \k ' Substituting the explicit values of the terms of the sequences and using the equality (3) at — ci\Q — qsB (r P \ psB — qrß G 71 + 1 Hr t a — ß \s q a.ia

n+

1 - 6i/3

n+

1 - -1- -a = 0, a-ß {at - aia)a

n - (bt - b

1ß)ß

r aa

n - bß

n (bt - btf)ß H

n aa

n - bß

r follows.