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

Approximation by quotients of terms of second order linear recursive 23 and let k and c be positive real numbers. Then with linear recurrences G(A

, B, qr, psB) and H(A, B

, 0, qsB) the inequality t G n + 1 H

r < 1 c\H

n\

k holds for infinitely many integer n if and only if (i) k < k

0 and c is arbitrary, (ii) or k = ko and c < CQ. (Note that k

0 > 0 since \B\ = \aß\ < a

2.) Corollary. Since t = 1- + ^c* is an irrational number, then G -'n + l Hr < cm holds with some c > 0 for infinitely many n if and only if \B\ = 1. 3. Examples 1

s

t Example, t = a is a real quadratic algebraic integer. Let G'(4,

19, Go, Gi), where Gq,G\ G Z not both GO and G'i are zero. The characteristic equation is x

2 — 4a; — 19 = 0 and a = (4 + \/92)/2. If approximation is done according to [3], the quality of approximation k

0 = 2 - = 0.4634845713 .. . The equation 92 = A

2 + 4B can be written in an infinite variety forms: ..., 2

2 + 4 • 22, 4

2 + 4 • 19, G

2 + 4 • 14, 8

2 + 4 • 7,

10

2 - 4 • 2,

12

2 - 4 • 13,... . Using IB\ of minimum value c*i = lo+^ioos = ß = Q e Q(ari), a = ax-3. G(10, -2, -3, -2), tf (10, -2, 0, -2) and thus k

0 = 2- = 1,696248791... . 2

n

d Example. / is a real quadratic non-algebraic integer. Let t be the root of larger absolute value of the equation 36x

2 — 894x- -f 1399 = 0. The roots of x

2 — 894a: + 36 • 1399 = x

2 - 894a;

+ 50364 = 0 are c*i and ß

1. Since t = ^ai, i.e. t G Q(»i ), we can approximate t. k

0 = 2- ^^L = 0, 3902074312 . . ., c

0 = 0, 002251014 .... Since D = 894

2-4-3G1399 = 2

2-3

4-(43

2-4), s/D = 2-3

2 V(

43

2 - 4)>

if

c follows that t G Q(a) is also true for the root a of x

2 — 43a; + 1 = 0. Indeed, t = | + ^cv and thus G'(43, —1, 10, —3), ii (43, —1, 0, —6). If we approximate a by the quotients Gn+i/Hn

) we get ko = 2, Co = 2, 386303511 . . ., and thus holds for infinitely many n. H n ^ cHl ^ H\