Numerical calculation of the density of prime numbers with a given least primitive root

On the least prime primitive root modulo a prime

We derive a conditional formula for the natural density E(q) of prime numbers p having its least prime primitive root equal to q, and compare theoretical results with the numerical evidence. 1. Theoretical result concerning the density of primes with a given least prime primitive root Let us denote, following Elliott and Murata [4], by g(p) and G(p) the least primitive and the least prime primi...

Quasi-Primitive Rings and Density Theory

A Novel Method of Searching Primitive Roots Modulo Fermat Prime Numbers

Primitive root is a fundamental concept in modern cryptography as well as in modern number theory. Fermat prime numbers have practical uses in several branches of number theory. As of today, there is no simple general way to compute the primitive roots of a given prime, though there exists methods to find a primitive root that are faster than simply trying every possible number. We prove the eq...

Ab initio study & Density functional investigational of Adenine & Thymine; Comparison of primitive Gaussian and NBO calculation

We have performed quantum-chemical ab initio in various basis sets at the Hartree-Fock andB3LYP levels for Adenine and Thymine with the program GAUSSIAN98. The Dipole,Quadrupole, Octapole and Hexadecapole moments and primitive in nine basis sets for thesemolecules are presented. The most stabilized forms of these molecules are observed in 6-311++G** basis set. We have evaluated coefficient hybr...

Least primitive root of prime numbers

Let p be a prime number. Fermat's little theorem [1] states that a^(p-1) mod p=1 (a hat (^) denotes exponentiation) for all integers a between 1 and p-1. A primitive root [1] of p is a number r such that any integer a between 1 and p-1 can be expressed by a=r^k mod p, with k a nonnegative integer smaller that p-1. If p is an odd prime number then r is a primitive root of p if and only if r^((p-...