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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
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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

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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