On a Three Dimensional Approximation Problem

Abstract

Let $\{ {R_n}\} \begin{array}{*{20}{c}} \infty \\ {n = 0} \\ \end{array}$ and $\{ {V_n}\} \begin{array}{*{20}{c}} \infty \\ {n = 0} \\ \end{array}$ be second order linear recurring sequences of integer defined by

$$ {R_n} = A{R_{n - 1}} - B{R_{n - 2}}{\rm{ }}(n > 1) $$

,

$$ {V_n} = A{V_{n - 1}} - B{V_{n - 2}}{\rm{ }}(n > 1) $$

where A > 0 and B are fixed non-zero integers and the initial terms of the sequences are R ₀ = 0, R ₁ = 1, V ₀ = 2 and V ₁ = A. Let α and β be the roots of the characteristic polynomials x ² = Ax + B of these sequences and denote its discriminant by D. Then, we have

$$ \sqrt D {\rm{ = }}\sqrt {{A^2} - 4B} {\rm{ }} = \alpha - \beta ,{\rm{ }}A = \alpha + \beta ,{\rm{ }}B = \alpha \beta $$

((1))

.