MathDB

We examine the following two sequences: The Fibonacci sequence:

F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }

for

n \geq 2

; The Lucas sequence:

L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}

for

n \geq 2

. It is known that for all

n \geq 0

F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},

where

\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}

. These formulae can be used without proof.
The coordinates of all vertices of a given rectangle are Fibonacci numbers. Suppose that the rectangle is not such that one of its vertices is on the

x

-axis and another on the

y

-axis. Prove that either the sides of the rectangle are parallel to the axes, or make an angle of

45^{\circ}

with the axes.

Problem Statement