Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T00:15:20.404Z Has data issue: false hasContentIssue false
  • English
  • Français

A parity problem from sieve theory

Published online by Cambridge University Press:  26 February 2010

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Magdalen College, Oxford, OX1 4AU
Get access

Extract

Let Ω(n) denote the number of prime factors of n, counted according to multiplicity. We shall consider the following question. Are there infinitely many natural numbers n for which Ω(n) = Ω(n + 1)? Erdős and Mirsky [4] have asked a closely related question concerning the divisor function d(n)—are there infinitely many n for which d(n) = d(n + 1)? The fact that Ω(n)is completely additive makes our problem slightly easier.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 1 - 6
Copyright
Copyright © University College London 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bombieri, E.. The asymptotic sieve. Rend. Accad. Naz. dei XL, V. Ser., 1-2 (1976), 243–269.Google Scholar
2.Chen, J.-R.On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica, 16 (1973), 157–176.Google Scholar
3.Elliott, P. D. T. A. and Halberstam, H.. A conjecture in prime number theory. Symposia Mathematika, 4 (INDAM, Rome, 1968/1969), 59–72.Google Scholar
4.Erdős, P. and Mirsky, L., The distribution of values of the divisor function d(n). Proc. London Math. Soc. (3), 2 (1952), 257–271.CrossRefGoogle Scholar
5.Halberstam, H. and Richert, H.-E.. Sieve methods (Academic Press, London, 1974).Google Scholar
6.Vaughan, R. C.. A remark on the divisor function d(n). Glasgow Math. J., 14 (1973), 54–55.Google Scholar