On simultaneous primitive roots

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 Commutation and Primitive Roots of Codes

Fibonacci Primitive Roots

(1) g 2 = g + 1 (mod p). It i s obvious that if (1) holds then so do (2) g 3 = g 2 + g (mod p) , (3) g 4 = g 3 + g 2 (mod p) , e t c .

Lucas Primitive Roots

is called the characteristic polynomial of the sequence U. In the case where P = -g = 1, the sequence U is the Fibonacci sequence and we denote its terms by F0, Fl9 F2, ... . Let p be an odd prime with p\Q and let e > 1 be an integer. The positive integer u = u(p) is called the rank of apparition of p in the sequence U if p\Uu and p\Um for 0 < m < u; furthermore, u = u(p) is called the period o...