Home Mathematics Bounded gaps between primes with a given primitive root, II
Article
Licensed
Unlicensed Requires Authentication

Bounded gaps between primes with a given primitive root, II

  • and EMAIL logo
Published/Copyright: July 15, 2015

Abstract

Let m be a natural number, and let 𝒬 be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q𝒬 as a primitive root, all lying in an interval of length O𝒬(exp(Cm)). This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin’s conjecture. Let E/ be an elliptic curve with an irrational 2-torsion point. Assume GRH. Then for every m, there are infinitely many strings of m consecutive primes p for which E(𝔽p) is cyclic, all lying an interval of length OE(exp(C′′m)). If E has CM, then the GRH assumption can be removed. Here C, C, and C′′ are absolute constants.

Award Identifier / Grant number: DMS-1402268

Funding statement: The second author thanks the NSF for their support under award DMS-1402268.

We are indebted to Pete L. Clark for helpful conversations on the theory of elliptic curves. This research began while the second author enjoyed a very pleasant visit to Brigham Young University. He thanks the BYU mathematics department for their hospitality.

References

[1] Akbary A. and Murty V. K., An analogue of the Siegel–Walfisz theorem for the cyclicity of CM elliptic curves mod p, Indian J. Pure Appl. Math. 41 (2010), 25–37. 10.1007/s13226-010-0002-4Search in Google Scholar

[2] Banks W. D., Freiberg T. and Turnage-Butterbaugh C. L., Consecutive primes in tuples, Acta Arith. 167 (2015), 261–266. 10.4064/aa167-3-4Search in Google Scholar

[3] Cassels J. W. S. and Fröhlich A., Algebraic Number Theory, Academic Press, London, 1986. Search in Google Scholar

[4] Clark P. L., Cook B. and Stankewicz J., Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice), Int. J. Number Theory 9 (2013), 447–479. 10.1142/S1793042112501436Search in Google Scholar

[5] Cojocaru A. C., Cyclicity of CM elliptic curves modulo p, Trans. Amer. Math. Soc. 355 (2003), 2651–2662, (electronic). 10.1090/S0002-9947-03-03283-5Search in Google Scholar

[6] Cojocaru A. C. and Murty. M. R., Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem, Math. Ann. 330 (2004), 601–625. 10.1007/s00208-004-0562-xSearch in Google Scholar

[7] Duke W., Almost all reductions modulo p of an elliptic curve have a large exponent, C. R. Math. Acad. Sci. Paris 337 (2003), 689–692. 10.1016/j.crma.2003.10.006Search in Google Scholar

[8] Gupta R. and Murty M. R., A remark on Artin’s conjecture, Invent. Math. 78 (1984), 127–130. 10.1007/BF01388719Search in Google Scholar

[9] Gupta R. and Murty M. R., Cyclicity and generation of points mod p on elliptic curves, Invent. Math. 101 (1990), 225–235. 10.1007/BF01231502Search in Google Scholar

[10] Gupta R., Murty M. R. and Murty V. K., The Euclidean algorithm for S-integers, Number Theory (Montreal 1985), CMS Conf. Proc. 7, American Mathematical Society, Providence (1987), 189–201. Search in Google Scholar

[11] Heath-Brown D. R., Artin’s conjecture for primitive roots, Quart. J. Math. Oxford (2) 37 (1986), 27–38. 10.1093/qmath/37.1.27Search in Google Scholar

[12] Hooley C., On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220. 10.1515/crll.1967.225.209Search in Google Scholar

[13] Lagarias J. C. and Odlyzko A. M., Effective versions of the Chebotarev density theorem, Algebraic Number Fields. L-Functions and Galois Properties (Durham 1975), Academic Press, London (1977), 409–464. Search in Google Scholar

[14] Lang S., Elliptic Functions, 2nd ed., Grad. Texts in Math. 112, Springer, New York, 1987. 10.1007/978-1-4612-4752-4Search in Google Scholar

[15] Li H. and Pan H., Bounded gaps between primes of the special form, Int. Math. Res. Not. IMRN (2015), 10.1093/imrn/rnv049. 10.1093/imrn/rnv049Search in Google Scholar

[16] Maynard J., Dense clusters of primes in subsets, preprint 2014, http://arxiv.org/abs/1405.2593. 10.1112/S0010437X16007296Search in Google Scholar

[17] Moree P., Artin’s primitive root conjecture—A survey, Integers 12 (2012), 1305–1416. 10.1515/integers-2012-0043Search in Google Scholar

[18] Murty M. R., On Artin’s conjecture, J. Number Theory 16 (1983), 147–168. 10.1016/0022-314X(83)90039-2Search in Google Scholar

[19] Murty M. R. and Murty V. K., A variant of the Bombieri–Vinogradov theorem, Number Theory (Montreal 1985), CMS Conf. Proc. 7, American Mathematical Society, Providence (1987), 243–272. Search in Google Scholar

[20] Murty M. R. and Srinivasan S., Some remarks on Artin’s conjecture, Canad. Math. Bull. 30 (1987), 80–85. 10.4153/CMB-1987-012-5Search in Google Scholar

[21] Pintz J., Are there arbitrarily long arithmetic progressions in the sequence of twin primes?, An Irregular Mind, Bolyai Soc. Math. Stud. 21, János Bolyai Mathematical Society, Budapest (2010), 525–559. 10.1007/978-3-642-14444-8_15Search in Google Scholar

[22] Pollack P., Bounded gaps between primes with a given primitive root, Algebra Number Theory 8 (2014), no. 7, 1769–1786. 10.2140/ant.2014.8.1769Search in Google Scholar

[23] Serre J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331. 10.1007/978-3-642-39816-2_94Search in Google Scholar

[24] Serre J.-P., Résumé des cours de l’année scolaire 1977–1978, Ann. Coll. France 1978 (1978), 67–70. Search in Google Scholar

[25] Serre J.-P., Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci. 54 (1981), 323–401. 10.1007/978-3-642-39816-2_125Search in Google Scholar

[26] Silverman J. H., The Arithmetic of Elliptic Curves, 2nd ed., Grad. Texts in Math. 106, Springer, New York, 2009. 10.1007/978-0-387-09494-6Search in Google Scholar

[27] Thorner J., Bounded gaps between primes in Chebotarev sets, Res. Math. Sci. 1 (2014), Article No. 4. 10.1186/2197-9847-1-4Search in Google Scholar

[28] Washington L. C., Introduction to Cyclotomic Fields, 2nd ed., Grad. Texts in Math. 83, Springer, New York, 1997. 10.1007/978-1-4612-1934-7Search in Google Scholar

Received: 2014-7-28
Published Online: 2015-7-15
Published in Print: 2016-7-1

© 2016 by De Gruyter

Downloaded on 22.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2014-0137/html
Scroll to top button