Two Symmetric Properties of Mersenne Numbers and Fermat Numbers

TL;DR

This paper introduces a new group construction for odd numbers, presents an algorithm to find the multiplicative order of 2 modulo q, and uncovers two symmetric properties of Mersenne and Fermat numbers.

Contribution

It proposes a novel group structure for odd numbers and reveals two symmetric properties of Mersenne and Fermat numbers, along with an algorithm for multiplicative order.

Findings

Discovery of two symmetric properties of Mersenne and Fermat numbers

Algorithm for determining the multiplicative order of 2 modulo q

Construction of a special group for each positive odd number

Abstract

Mersenne numbers and Fermat numbers are two hot and difficult issues in number theory. This paper constructs a special group for every positive odd number other than 1, and discovers an algorithm for determining the multiplicative order of 2 modulo q for each positive odd number q. It is worth mentioning that this paper discovers two symmetric properties of Mersenne numbers and Fermat numbers.