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

Abstract

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

1

(g, f), p

2

(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

j

(g, f) with 1 ≤ j ≤ r. We let c

g

(f) be the number of

distinct primes amongst p

j

(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

g

(f) is as larg e as possible a nd it was stated that

c

g

(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

c

g

(f) = 38639.

It is obtained on taking f (X) = 32X

2

+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

2

+ 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

g

(f) = 37951. This improves on the record indicated in

[6], but falls slightly below the ‘hidden record’ indicated a bove.

1

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

m

(1). For related work see Baker and Pollack [2].

References

[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–

220.

[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,

arXiv:1404.4007v3.

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