The remainders of the Euclidean algorithm with symmetric quotients

TL;DR

This paper derives explicit formulas for remainders in the Euclidean algorithm when quotients are symmetric, enabling analysis of related quadratic forms and deepening understanding of the algorithm's structure.

Contribution

It introduces explicit formulas for remainders in symmetric quotient cases, facilitating analysis of quadratic forms linked to the Euclidean algorithm.

Findings

Formulas for remainders with symmetric quotients

Analysis of quadratic forms in remainders

Enhanced understanding of Euclidean algorithm structure

Abstract

When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.