A Survey Review Report on Primitive Roots and Power Residues

Tannu Gupta, Dr. Manu Gupta

Abstract: The concepts of primitive roots and power residues play a fundamental role in the structure and behavior of number systems, particularly within the field of modular arithmetic. This research paper explores the theoretical foundations and mathematical significance of primitive roots integers that generate the multiplicative group of integers modulo n and their close relationship to power residues, which are the possible values of integer powers modulo a given number. Through a rigorous study of these concepts, the paper highlights their applications in cryptography, coding theory, primality testing, and advanced algebraic structures. By examining various theorems, such as Euler's theorem and Gauss's work on residues, this study provides insight into the distribution and behavior of primitive roots and residues in different modulo systems. Moreover, practical examples and computational analysis are used to illustrate their importance in both pure and applied mathematics. This research aims to deepen understanding and encourage further exploration in areas where number theory serves as the foundation for modern mathematical applications.

Keywords: Primitive Roots, Power Residues, Modular Arithmetic, Number Theory, Euler's Theorem, Cyclic Groups, Cryptography, Fermat's Little Theorem, Residue Classes, Discrete Logarithm, Primality Testing, Modular Exponentiation, Algebraic Number Theory, Multiplicative Order, Finite Fields

How to Cite?: Tannu Gupta, Dr. Manu Gupta, "A Survey Review Report on Primitive Roots and Power Residues", Volume 14 Issue 7, July 2025, International Journal of Science and Research (IJSR), Pages: 1786-1788, https://www.ijsr.net/getabstract.php?paperid=SR25729120933, DOI: https://dx.doi.org/10.21275/SR25729120933