Abstract
The Green–Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu’s Theorem to certain subsets of the primes such as primes of the form \({\lfloor{\pi n}\rfloor}\) and some of arithmetic density zero such as primes of the form \({\lfloor{n\log\log n}\rfloor}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banks W.D., Shparlinski I.E.: Prime numbers with Beatty sequences. Colloquium Mathematicum 115, 147–157 (2009)
De Bruijn N.G.: On the number of positive integers \({\leq x}\) and free of prime factors \({\geq y}\) . Indag. Math. 13, 50–60 (1951)
A. Granville, Unexpected Irregularities in the Distribution of Prime Numbers, Proceedings of the ICM, 1994.
Green B., Tao T.: The Primes Contain Arbitrarily Long Arithmetic Progressions. Ann. Math, 167, 481–547 (2008)
Gowers W.T.: A New Proof of Szemerédi’s Theorem, GAFA. Geom. funct. anal. 11, 465–588 (2001)
G. H. Hardy, and E. M. Wright, An Introduction to the Theory of Numbers, D. R. Heath-Brown and J. H. Silverman, eds., 6th ed., Oxford University Press, Oxford, 2008.
Khinchin A.Y.: Zur metrischen Theorie der diophantischen Approximationen. Math. Z. 24, 706–714 (1926)
Kra B.: The Green–Tao Theorem on Arithmetic Progressions in the Primes: An Ergodic Point of View. Bull. Amer. Math. Soc. 43, 3–23 (2006)
Leitmann D.: The Distribution of Prime Numbers in Sequences of the Form \({\lfloor f(n)\rfloor}\) , Proc. London Math. Soc. 35, 448–462 (1977)
Maier H.: Chains of large gaps between consecutive primes. Adv. Math. 39, 257–269 (1981)
Roth K.F.: Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955)
Roth K.F.: Corrigendum to “Rational approximations to algebraic numbers”. Mathematika 2, 168 (1955)
Shiu D.K.L.: Strings of Congruent Primes. J. London Math. Soc. 61, 359–373 (2000)
F. Thorne, Extensions of Results on the Distribution of Primes, ProQuest, UMI Dissertation Publishing (2011).
van der Corput J.G.: Über Summen von Primzahlen und Primzahlquadraten. Math. Ann. 116, 1–50 (1939)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are grateful to the NSF’s support of the REU at Emory University.
Rights and permissions
About this article
Cite this article
Monks, K., Peluse, S. & Ye, L. Strings of special primes in arithmetic progressions. Arch. Math. 101, 219–234 (2013). https://doi.org/10.1007/s00013-013-0544-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-013-0544-x