New algorithms for modular inversion and representation by binary quadratic forms arising from structure in the Euclidean algorithm

TL;DR

This paper introduces new algorithms for computing modular inverses and representing primes by specific binary quadratic forms, leveraging structured patterns in the Euclidean algorithm's quotient and remainder sequences.

Contribution

It presents novel algorithms based on Euclidean algorithm structures for modular inversion and quadratic form representation, with specific input families.

Findings

New algorithms for modular inversion

Representation of primes by quadratic forms

Utilization of Euclidean algorithm structure

Abstract

We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by the binary quadratic form $x^2 + 3xy + y^2$. The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.