The Pell equation over the Fibonacci ∘-ring

Abstract

The paper considers the Pell equation

$$N_1 \circ N_1 - A \circ N_2 \circ N_2 = 1$$

over the Fibonacci ∘-ring $\mathop \mathbb{Z}\limits^ \circ$, which is obtained by supplying the ring of integers ℤ with the Fibonacci circle multiplication operation N ∘ M. It is proved that if a positive integer A satisfies the condition Aτ < [(A + 1)τ], where $\tau = \tfrac{{ - 1 + \sqrt 5 }}{2}$ is the golden section, and [x] is the integral part of x, then the Pell equation is solvable both in integers and in positive integers N₁ and N₂. Moreover, for the number n(A; X) of integer solutions (N₁, N₂), |N₁| ≤ X, lower bounds are established. Bibliography: 7 titles.