Modular Inverse and Reciprocity Formula

TL;DR

This paper introduces a reciprocity formula for modular inverses of integers and Gaussian integers, providing a new tool for calculating and verifying modular inverses in number theory applications.

Contribution

It presents a novel reciprocity formula for modular inverses and explores its applications in computing inverses of Gaussian integers.

Findings

Proves a new reciprocity formula for modular inverses.

Demonstrates applications in calculating inverses of Gaussian integers.

Provides methods for verifying specific modular inverses.

Abstract

This paper proves a reciprocity formula for modular inverses for non-zero integers and demonstrates some applications of the reciprocity formula in calculating or verifying some modular inverses of specific forms, including the modular inverse of a Gaussian integer modulo another Gaussian integer.