Philippine E-Journals

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

Ricky B. Villeta | Elmer C. Castillano | Roberto N. Padua

 

Abstract:

This paper further investigates the cyclic group (Zp) with respect to the primitive roots or generators g ∈ (Zp). The simulation algorithm that determines the generators and the number of generators, g of (Zp) for a prime p is illustrated using Python programming. The probability of getting a generator g of (Zp) denoted by, φ(φ(p))/ φ(p) is generated for prime p between 0 to 3000. The scatterplot is also shown that depicts the data points on the probability φ(φ(p))/ φ(p) 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 (Zp) 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.