On indices modulo a number with no primitive roots
Date of Publication
1995
Document Type
Bachelor's Thesis
Degree Name
Bachelor of Science in Mathematics
College
College of Science
Department/Unit
Mathematics and Statistics
Abstract/Summary
A system of linear simultaneous congruences is a system of congruences that involves only one variable, but different moduli. The process of obtaining a solution to this system has a long history, appearing in the Chinese literature as early as the first century A.D.. Such systems arose in ancient Chinese puzzles such as the following: Find a number that leaves a remainder of 1 when divided by 3, a remainder of 2 when divided by 5, and a remainder of 3 when divided by 7.The rule for obtaining a solution goes by the name of the Chinese Remainder Theorem. An application of the Chinese Remainder Theorem leads to the main theorem of this paper which gives an easier way in finding indices modulo a number n which admits no primitive roots.The main theorem assumes that the integers n1, nj satisfy gcd (ni, nj) = 1, when I j where ni 2 (i = 2, ..., s) and n = n1 n2 ....n s. It also assumes that each n i admits a primitive root r i. We let x i denote the solutions of the s Chinese Remainder problems x1 = r1 (mod n1) x2 = 1 (mod n1) ... xs = 1 (mod n1) x1 = 1 (mod n2) x2 = r2 (mod n2) ... xs = 1 (mod n2) x1 = 1 (mod ns) x2 = 1 (mod ns) ... xs = r s (mod ns). Then every element M of a reduced residue system modulo n is produced exactly once by the congruences, M = k1 x1 X k2 x2 ... xs ks (mod n) where k i = 0, 1, 2, ... (ni) - 1, (i = 1, 2, ..., s).
Abstract Format
html
Language
English
Format
Accession Number
TU07062
Shelf Location
Archives, The Learning Commons, 12F, Henry Sy Sr. Hall
Physical Description
100 leaves
Keywords
Number theory of; Linear systems; Congruences and residues; Groups, Theory of; Diophantine analysis
Recommended Citation
Medina, M. L., & Mendoza, M. O. (1995). On indices modulo a number with no primitive roots. Retrieved from https://animorepository.dlsu.edu.ph/etd_bachelors/16261