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 s(a,b;X) of points 1,x2)
in the two-dimensional Fibonacci quasilattices m2 of
level ¼ lying on the
hyperbola 12
- a
x22 = b and such that
£
x1 £
X, 2 ³
0, the asymptotic formula
s(a,b;X)
~ cs(a,b) ln X as
X ®
¥
is established, the coefficient s(a,b) is
calculated exactly.
Using this, the following result is obtained.
Let m be the
Fibonacci numbers, i
Î N,
, and let
be the shift of i in the
Fibonacci numeral system.
Then the number s(X) of all
solutions 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.
$$
£
A1 £
X, 2 ³
0, satisfies the asymptotic formula
n_s(X) \sim \frac{c_s}{\mathrm{arcosh}(1/\tau)} \ln X \quad
\text{as}\quad X \rightarrow \infty.
$$
Here t =
(-1 + √(5))/2
is the golden ratio, and
s = 1/2 or
for
or
³
1, respectively.
Location: http://mech.math.msu.su/~fpm/eng/k10/k106/k10605h.htm
Last modified: July 5, 2011