Artin prime producing quadratics

Abstract

Fix an integer g. The primes p such that g is a primitive root for p are called Artin primes. Using a mixture of heuristics, well-known conjectures and rigorous arguments an algorithm is given to find quadratics that produce many Artin primes. Using this algorithm Y. GALLOT has found a g and a quadratic f such that the first 31082 primes produced by f have g as a primitive root. There is a connection with finding integers d such that L (2, (d/)) is small.

Addendum to the paper: “Artin prime producing quadratics”, by P. Moree

“…This improves on the record indicated in [6], but falls slightly below the 'hidden record' indicated above.…”

“…In [6] the problem was addressed of finding an integer g and a quadratic polynomial f such that c g (f ) is as large as possible and it was stated that It is obtained on taking f (X) = 32X 2 +39721664X +182215381147285848449 and g = 593856338459898. Perhaps a more elegant reformulation is: for those 38639 integers n in [620651, 1749283] for which h(n) := 32n 2 + 182215368820640606817 is prime, the number 593856338459898 is a primitive root modulo h(n).…”

Artin's Primitive Root Conjecture – A Survey

“…D. Lehmer [288] found in 1963 that 326 is a primitive root for the first 206 primes of the form 326X 2 + 3. More impressive examples in the same spirit can be given using recent results on prime producing quadratics by Moree [352] and K. Scholten [451]. Y. Gallot holds the record in which 206 is being replaced by 38639.…”

Bounded gaps between primes with a given primitive root

“…Quantitative conjectures of this kind, but in the context of primes represented by a single irreducible polynomial rather than primes produced by linear forms, appear in recent work of Moree [11] and of Akbary and Scholten [1].…”