Bounded gaps between primes with a given primitive root

Abstract

Abstract. Fix an integer g = −1 that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which g is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard-Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer m ≥ 2. If q1 < q2 < q3 < . . . is the sequence of primes possessing g as a primitive root, then lim infn→∞(q n+(m−1) − qn) ≤ Cm, where Cm is a finite constant that depends on m but not on g. We also show that the primes qn, qn+1, . . . , qn+m−1 in this result may be taken to be consecutive.

Bounded gaps between prime polynomials with a given primitive root

“…This proof follows essentially the same strategy as Section 3.3 of [Pol14]. Sincẽ S 1 = S 1 , we need only concern ourselves withS 2 .…”

“…For the present article, we fix g satisfying the conditions of Theorem 1.3 and modify the above argument as necessary; our modifications are somewhat similar to those in [Pol14]. Given an admissible k-tuple H = {h 1 , .…”

“…Artin's famous primitive root conjecture states that for any integer g = −1 and not a square, there are infinitely many primes for which g is a primitive root; that is, there are infinitely many primes p for which g generates (Z/pZ) * . Work of Hooley [Hoo67] establishes the truth of Artin's conjecture, assuming GRH; the following result due to Pollack [Pol14] is a bounded gaps result in this setting.…”

Gaps Between Primes

“…The fact that one can restrict the entire argument to an arithmetic progression also allows one to get some control on the joint distribution of various arithmetic functions. There have been many recent works making use of these flexibilities in the setup of the sieve method, including [58,13,7,21,48,34,3,39,4,14,61,59,46,47,28,5,6,1,32,43,49].…”

A Novel Method of Searching Primitive Roots Modulo Fermat Prime Numbers

“…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 [10][11]. If m is a primitive root modulo p, then the multiplicative order of m is , where is Euler's totient function.…”