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Journal ArticleDOI

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

24 Feb 2015-Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg (Springer Berlin Heidelberg)-Vol. 85, Iss: 1, pp 87-88

AbstractA record mentioned in the paper by Moree (Abh Math Sem Univ Hamburg 77:109–127, 2007) was recently improved on by Akbary and Scholten. However, the record mentioned was not the then record. The then record, due to Gallot (2004), actually slightly improves on that obtained recently by Akbary and Scholten.

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  • The record mentioned was not the then record.
  • This improves on the record indicated in [6], but falls slightly below the ‘hidden record’ indicated above.

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arXiv:1501.02350v1 [math.NT] 10 Jan 2015
Addendum to the paper: “Artin Prime
Producing Quadratics” [Abh. Math. Sem. Univ.
Hamburg 77 (2007), 109–127; MR2379332
(2008m:11194)] by P. Moree
Yves Gallot and Pieter Moree
A record mentioned in the paper was recently improved on by Akbary
and Scholten. However, the record mentioned was not the then record.
The then record, due to Gallot (2004), actually slightly improves on that
obtained recently by Akbary and Scholten.
Given an integer g and a polynomial f(X) Z[X], let p
(g, f), p
(g, f), . . . be the
sequence of primes that is obtained on going through the sequence f (0), f (1), . . .
and writing down the primes not dividing g as they appear (called Artin primes).
We let r be the la r gest integer r (if this exists) such that g is a primitive root
mod p for all primes p
(g, f) with 1 j r. We let c
(f) be the number of
distinct primes amongst p
(g, f) with 1 j r.
In [6] the problem was addressed of finding an integer g and a quadratic
polynomial f such that c
(f) is as larg e as possible a nd it was stated that
(f) = 31082
was the current record (obtained by Yves Ga llot). On preparing the paper for
publication (fall 2006) the author failed to recall an e-mail by Gallot from June
2004. That e-mail actually stated what in 2006 still would be the t r ue current
record (due to Gallot), namely
(f) = 38639.
It is obtained on taking f (X) = 32X
+39721664X +182215381147285848449
and g = 593856338459898. Perhaps a mo r e elegant reformulation is: for those
38639 integers n in [620651, 1749283] for which
h(n) := 32n
+ 182215368820640606817
is prime, the number 593856338459898 is a primitive root modulo h(n).
In a recent paper by Akbary and Scholten [1] the authors find a g and a
quadratic f such that c
(f) = 37951. This improves on the record indicated in
[6], but falls slightly below the ‘hidden record’ indicated a bove.

Akbary a nd Scholten go beyond Moree in that t hey in addition consider the
case where f linear and f cubic and obtain here record values for consecutive
Artin primes for certain integers g of 6355, respectively 10011.
Finally, let us mention some highly interesting work by Pollack [7]. He merges
the method of proof of Hooley [3] of Artin’s conjecture (under GRH) with the
method of Maynard-Tao [4, 5] in order to produce bounded gaps between primes:
On GRH for every nonsquare g 6= 1 and every m, there are infinitely many
runs of m consecutive primes all possessing g as a primitive root and lying in an
interval of order O
(1). For related work see Baker and Pollack [2].
[1] A. Akbary a nd K. Scholten, Artin prime producing polynomials,
arXiv:1310.5198, to appear in Mathematics of Computation. (Electronically
published in December 2014.)
[2] R.C. Baker and P. Pollack, Bounded gaps between primes with a given prim-
itive root, II, arXiv:1407.7186.
[3] C. Hooley, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–
[4] J. Mayna r d, Small gaps between primes, Ann. Math., to appear .
[5] D.H.J. Polymath, Variants of the Selberg sieve, and bounded intervals con-
taining many primes, arXiv:1407.4897.
[6] P. Moree, Artin prime producing quadratics, arXiv:1407.4897. Abh. Math.
Sem. Univ. Hamburg 77 (2007), 109–127.
[7] P. Pollack, Bounded gaps between primes with a given primitive root,
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