On the order of odd integers modulo 2n

Soon-Mo Jung,Doyun Nam,Michael Rassias

doi:10.2298/aadm190326023j

Abstract

In this paper, we investigate the order of odd integers of the forms 2ju + 1 and 2ju+3 modulo 2n, where j is an integer with j ? 2, u is an odd positive integer, and n is an integer with n ? j + 3.

Highlights

The concept of divisibility seems intuitive, it plays a fundamental role in number theory and it leads to a number of foundational concepts such as the greatest common divisor, the least common multiple, and prime numbers
If n is an integer larger than 2, 2n does not have a primitive root. This means that the order of any odd integer modulo 2n is less than φ(2n), where φ stands for Euler’s totient function
The following theorem provides a lower bound for the order of odd integers of the form 2ju + 1, where j ≥ 2 and u is odd

Summary

Introduction

AND PRELIMINARIES the concept of divisibility seems intuitive, it plays a fundamental role in number theory and it leads to a number of foundational concepts such as the greatest common divisor, the least common multiple, and prime numbers. These facts imply that if n ≥ 3, the order of odd integers modulo 2n is less than φ(2n). In [6, Theorem 9.11], the upper bound for the order of odd integers modulo 2n is provided.

Results

Conclusion