FPM 2010, vol. 16, no. 6, pp. 45-62

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

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For the number $n$ _s(a,b;X) of points $(x$ 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) ~ c_s(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 $$ \overleftarrow {A}_i
$$ be the shift of $A$ _i in the Fibonacci numeral system. Then the number $n$ _s(X) of all solutions $(A$ 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.

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