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3-tuples have at most 7 prime factors infinitely often

Published online by Cambridge University Press:  18 June 2013

JAMES MAYNARD
JAMES MAYNARD*
Affiliation:
Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB. email: [email protected]
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Abstract

Let L1, L2L3 be integer linear functions with no fixed prime divisor. We show there are infinitely many n for which the product L1(n)L2(n)L3(n) has at most 7 prime factors, improving a result of Porter from 1972. We do this by means of a weighted sieve based upon the Diamond-Halberstam-Richert multidimensional sieve.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 443 - 457
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Chen, J. R.On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16 (1973), 157–176.Google Scholar
[2]Diamond, H. G. and Halberstam, H.A Higher-Dimensional Sieve Method. Cambridge Tracts in Math. no. 177 (Cambridge University Press, 2008). With an appendix (“Procedures for computing sieve functions”) by William F. Galway.CrossRefGoogle Scholar
[3]Heath-Brown, D. R.Almost-prime k-tuples. Mathematika 44 2 (1997), 245–266.CrossRefGoogle Scholar
[4]Ho, K.-H. and Tsang, K.-M.On almost prime k-tuples. J. Number Theory 120 1 (2006), 33–46.CrossRefGoogle Scholar
[5]Maynard, J. Almost-prime k-tuples. Pre-print, available at http://arxiv.org/abs/1205.4610v1.Google Scholar
[6]Montgomery, H. L. and Vaughan, R. C.Multiplicative Number Theory, I. Classical Theory. Cambridge Studies in Advanced Math. no. 97 (Cambridge University Press, 2007).Google Scholar
[7]Motohashi, Y.An induction principle for the generalization of Bombieri's prime number theorem. Proc. Japan Acad. 52 6 (1976), 273–275.Google Scholar
[8]Porter, J. W.Some numerical results in the Selberg sieve method. Acta Arith. 20 (1972), 417–421.CrossRefGoogle Scholar