On the Generators of the Group of Units Modulo a Prime and Its Analytic and Probabilistic Views

Abstract

This paper further investigates the cyclic group ( ) ∗ p Z with respect to the primitive roots or generators ( ) ∗ ∈ p Z g . The simulation algorithm that determines the generators and the number of generators, g of ( ) ∗ p Z for a prime p is illustrated using Python programming. The probability of getting a generator g of ( ) ∗ p Z , denoted by , is generated for prime p between 0 to 3000. The scatterplot is also shown that depicts the data points on the probability of the group of units with respect to the order p - 1 of for prime p between 0 to 3000. The scatterplot results reveal that the probability of getting a generator of the group of units is fluctuating within the probability range of 0.20 to 0.50, for prime p modulus from 3 to 3000. These findings suggest that the proportion of the number of generators of the group of units modulo a prime of order p - 1 , though fluctuating, is bounded from 20% to 50% for prime p modulus from 3 to 3000.