On prime primitive roots

Constructing the Primitive Roots of Prime Powers

We use only addition and multiplication to construct the primitive roots of p from the primitive roots of p, where p is an odd prime and k ≥ 2.

On simultaneous primitive roots

Given finitely many non zero rational numbers which are not ±1, we prove, under the assumption of Hypothesis H of Schinzel, necessary and sufficient conditions for the existence of infinitely many primes modulo which all the given numbers are simultaneously primitive roots. A stronger result where the density of the primes in consideration was computed was proved under the assumption of the Gen...

Notes on primitive lambda-roots

Euler’s totient function φ has the property that φ(n) is the order of the group U(n) of units in Zn (the integers mod n). In the early years of the twentieth century, Carmichael defined a similar function λ, where λ(n) is the exponent of U(n). He called an element of U(n) with order λ(n) a primitive λ-root of n. Subsequently, primitive λ-roots have not received much attention until recently, wh...

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...

On Commutation and Primitive Roots of Codes