# On Artin's Conjecture for Primitive Roots

@inproceedings{Pappalardi1993OnAC, title={On Artin's Conjecture for Primitive Roots}, author={Francesco Pappalardi}, year={1993} }

Various generalizations of the Artin’s Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field Q(ζl, 2 ) valid for the range l < log x is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary…

## 128 Citations

On a Generalization of Artin's Conjecture

- Mathematics
- 2008

A primitive root mod p is a generator of (Z/pZ)∗. Gauss was the first to introduce the idea of a primitive root in his Disquisitiones Arithmeticae. They were first used to answer questions about the…

Artin's Conjecture on Average for Composite Moduli

- Mathematics
- 2000

Let a be an integer ≠−1 and not a square. Let Pa(x) be the number of primes up to x for which a is a primitive root. Goldfeld and Stephens proved that the average value of Pa(x) is about a constant…

Artin's Conjecture: Unconditional Approach and Elliptic Analogue

- Mathematics
- 2008

In this thesis, I have explored the different approaches towards proving Artin’s ‘primitive root’ conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by…

A PROBLEM OF FOMENKO'S RELATED TO ARTIN'S CONJECTURE

- Mathematics
- 2012

Let a be a natural number greater than 1. For each prime p, let ia(p) denote the index of the group generated by a in . Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko…

Orders of algebraic numbers in finite fields

- Mathematics
- 2021

For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the…

The distribution of sequences in residue classes

- Mathematics
- 2002

We prove that any set of integers A C [1,x] with |A| » (log x) r lies in at least v A (p) » p r/r+1 many residue classes modulo most primes p « (log x) r+1 . (Here r is a positive constant.) This…

NEW OBSERVATIONS ON PRIMITIVE ROOTS MODULO PRIMES

- Mathematics, Computer Science
- 2020

A theorem concerning ∑ g( g+c p ), where g runs over all the primitive roots modulo p among 1, .

Note on Artin's Conjecture on Primitive Roots

- MathematicsHardy-Ramanujan Journal
- 2022

E. Artin conjectured that any integer $a > 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square…

A remark on the Lang-Trotter and Artin conjectures

- Mathematics
- 2018

We use recent advances in sieve theory to show that conditional upon the generalized Elliott-Halberstam conjecture, at least one of the following is true: (i) Artin’s primitive root conjecture holds…

On Primes Recognizable in Deterministic Polynomial Time

- Mathematics, Computer ScienceThe Mathematics of Paul Erdős I
- 2013

It is shown that it is easy to find a proof of primality for a prime p if the complete factorization of p - 1 is known, and every prime p can be deterministically supplied with aProof of its primality in O((logp)3) arithmetic steps with integers at most p.

## References

SHOWING 1-10 OF 67 REFERENCES

On Artin's conjecture.

- Mathematics
- 1967

The problem of determining the prime numbers p for which a given number a is a primitive root, modulo JP, is mentioned, for the partieular case a — 10, by Gauss in the section of the Disquisitiones…

Counting Points Modulo p for some Finitely Generated Subgroups of Algebraic Groups

- Mathematics
- 1982

We begin by explaining the basic idea of this paper in a simple case. We write n p for the order of 2 modulo the prime p, so that n p is the number of powers of 2 which are distinct mod p. We have…

Cyclicity and generation of points mod p on elliptic curves

- Mathematics
- 1990

In this paper we study the group of points modulo p of elliptic curves defined over Q. In particular, we are interested in the frequency with which this group is cyclic and with which it is generated…

A remark on Artin's conjecture

- Mathematics
- 1984

A famous conjecture o f E. Ar t in [ t ] s tates that for any integer a 4= +_ I or a perfect square, there are infini tely many pr imes p for which a is a pr imit ive roo t (modp) . This conjecture…

Algebraic Number Theory

- Mathematics
- 1971

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary…

Multiplicative Number Theory

- Mathematics
- 1967

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The…

Artin’s conjecture for primitive roots

- Mathematics
- 1988

In his preface to Diophantische Approximationen, Hermann Minkowski expressed the conviction that the “deepest interrelationships in analysis are of an arithmetical nature.” Gauss described one such…

The Riemann Zeta-Function

- Mathematics
- 1992

In order to make progress in number theory, it is sometimes necessary to use techniques from other areas of mathematics, such as algebra, analysis or geometry. In this chapter we give some…

On the Sieve of Eratosthenes

- MathematicsCanadian Journal of Mathematics
- 1987

Let v(n) denote the number of distinct prime factors of a natural number n. A classical theorem of Hardy and Ramanujan states that the normal order of v(n) is log log n. That is, given any , the…