scispace - formally typeset

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.

Summary (1 min read)

Jump to:  and [Summary]

Summary

  • 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.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

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.
2
References
More filters

Journal Article
Abstract: 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 Arithmeticae that is devoted to the periodie deeimal expansions of fraetions with denominator p. Several writers in the nineteenth Century subsequently alluded to the problem, but, since their results were for the most part of an inconclusive nature, we are content to single out from their work the interesting theorem that 2 is a primitive root, modulo p, if p be of the form 4g + l, where q is a prime. The early work, however, was eonfined almost entirely to special cases, it not being until 1927 that the problem was formulated definitely in a general sense. In the latter year the late Emil Artin enunciated the celebrated hypothesis, now usually known äs Artin's conjecture, that for any given non-zero integer a other than l, —l, or a perfect square there exist infinitely many primes p for which is a primitive root, modulo p. Furthermore, letting Na(#) denote the number of such primes up to the limit #, he was led to conjecture an asymptotic formula of the form

413 citations


Journal ArticleDOI
Abstract: Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these polynomials is irreducible over the field of rational numbers and no two of them differ by a constant factor. Let Q(fi , f2, ... , fk ; N) denote the number of positive integers n between 1 and IV inclusive such that fi(n) , f2(n), , fk(n) are all primes. (We ignore the finitely many values of n for which some fi(n) is negative.) Then heuristically we would expect to have for N large

272 citations


Journal ArticleDOI
Abstract: We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m pn) <1 for every integer m. We also show that lim inf(pn+1 pn) 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 pn) 12 and lim infn(pn+2 pn) 600.

271 citations


Journal ArticleDOI
Abstract: For any m≥1, let H m denote the quantity . A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H m for m≥2, and specifically that H m ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yildirim, as well as the bound H m ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound or H m ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H m when m=2,3,4,5.

130 citations


Journal ArticleDOI
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

36 citations