Polynomial sequences on quadratic curves

TL;DR

This paper generalizes the study of integer points on conics by introducing radical points, linking solutions of Diophantine equations to recurrence sequences and special polynomials, revealing new identities and relationships.

Contribution

It introduces radical points on quadratic curves, connecting solutions of Diophantine equations with recurrence sequences and classical polynomials, expanding previous work on conics.

Findings

Solutions to certain Diophantine equations are expressed via linear recurrence sequences.

Identifies relationships between these sequences and known OEIS sequences.

Establishes connections between these sequences and Chebyshev and Morgan-Voyce polynomials.

Abstract

In this paper we generalize the study of Matiyasevich on integer points over conics, introducing the more general concept of radical points. With this generalization we are able to solve in positive integers some Diophantine equations, relating these solutions by means of particular linear recurrence sequences. We point out interesting relationships between these sequences and known sequences in OEIS. We finally show connections between these sequences and Chebyshev and Morgan-Voyce polynomials, finding new identities.