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

22 S. H.-Molnár holds for infinitely many integer n if and only if (i) k < ko and c is arbitrary, (ii) or k = ko and c < Co, (iii) or k = ko, c = Co and B > 0, (iv) or k — ko, c = Co, B < 0 and b/a > 0, G

n < where k

0 = 2 - and c

0 - |

a|

fc

o-i|

&

r If D < 0 then a and ß are non real complex numbers with |a| = \ß\ and by (1) we have = • But \ß/a\ = 1, thus Hm does not even exist. The approximation of |tv| by rationals of the form |G

n-|-i/G

n| was considered e.g. in [3], [4] and [5]. In [3] it was proved that if G is a non-degenerate second order linear recurrence with D < 0 and initial values Go = 0, G'i = 1, then there exists a constant ci > 0, depending only on the sequence G, such that for infinitely many n. In this paper the root a of the characteristic polynom of the sequence G will not be approximated by the quotients G

n+\/G

n

, but by G

n+i/H

n, where H is an appropriately chosen second order linear recursive sequence. We can always give a better approximation for |cv| if D < 0, and for a in the most cases if D > 0 as it was given by the authors in [3]. This can be achieved by the approximation of the numbers of the quadratic number field Q(a) when D > 0. The theorems in [3] can only approximate quadratic algebraic integers. Since at least one real quadratic algebraic integer a can be found for any real quadratic algebraic number 7, such that 7 6 Q (a

), our theorem can adequately approximate any irrational quadratic algebraic number, independently whether it is an algebraic integer or not. We are going to illustrate the above statement and its applicability to non-real complex quadratic algebraic numbers. 2. Result We prove the following theorem: Theorem. Let A and B be rational integers with the restrictions AB / 0 and D = A

2+4B > 0 is not a perfect square. Denote by a and ß the roots of equation x

2 — Ax — B = 0, where |c*| > \ß\. Let t - ~ + £ Q(a) with integers s, q > 0,p / 0 and r. Dehne the numbers k

0 and Cq by k 0 _ 2 — log Iff 1 log H and CQ = y/D qsB h o-l 1