FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 6, PAGES 45-62

## Hyperbolas over two-dimensional Fibonacci quasilattices

V. G. Zhuravlev

Abstract

View as HTML     View as gif image

For the number $n$s(a,b;X) of points $\left(x$1,x2) in the two-dimensional Fibonacci quasilattices $F$m2 of level $m = 0,1,2,$¼ lying on the hyperbola $x$12 - a x22 = b and such that $0$£ x1 £ X, $x$2 ³ 0, the asymptotic formula

$n$s(a,b;X) ~ cs(a,b) ln X as X ® ¥

is established, the coefficient $c$s(a,b) is calculated exactly. Using this, the following result is obtained. Let $F$m be the Fibonacci numbers, $A$i Î N, $i = 1,2$, and let $\overleftarrow \left\{A\right\}_i$ be the shift of $A$i in the Fibonacci numeral system. Then the number $n$s(X) of all solutions $\left(A$1,A2) of the Diophantine system


\left\{ \begin{aligned}
& A_1^2+\overleftarrow {A}_1^2- 2 A_2 \overleftarrow {A}_2+ \overleftarrow {A}_2^2 = F_{2s},
& \overleftarrow {A}_1^2- 2 A_1 \overleftarrow {A}_1+A_2^2 - 2 A_2 \overleftarrow {A}_2+ 2 \overleftarrow {A}_2^2 = F_{2s-1},
\end{aligned} \right.
$$$0$£ A1 £ X, $A$2 ³ 0, satisfies the asymptotic formula  n_s(X) \sim \frac{c_s}{\mathrm{arcosh}(1/\tau)} \ln X \quad \text{as}\quad X \rightarrow \infty.$$

Here t = (-1 + √(5))/2 is the golden ratio, and $c$s = 1/2 or $1$ for $s = 0$ or $s$³ 1, respectively.

Location: http://mech.math.msu.su/~fpm/eng/k10/k106/k10605h.htm