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Heath-Brown, D. R.
- A Note on the Paper ‘Consecutive Almost-Primes’
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d1,d2∈S, d1<d2⇒d2-d1|d1,d2.
Authors
Affiliations
1 Magdalen College, Oxford, GB
1 Magdalen College, Oxford, GB
Source
The Journal of the Indian Mathematical Society, Vol 66, No 1-4 (1999), Pagination: 203-205Abstract
The author’s paper [1], quoted in the title, made important use of so-called ‘special sets’ S⊆N, with the defining property thatd1,d2∈S, d1<d2⇒d2-d1|d1,d2.
- Consecutive Almost-Primes
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P(ϵ)={n∈N: n=pk, p prime, k≤n}.
Thus if ϵ is small, the elements of P(ϵ) are “almost prime”. (Note that this phrase is used here in a different sense from that of the weighted sieve, in which n=Pr is an almost-prime if n has at most r prime factors.) It was conjectured by Erdos [1] that for any ϵ>0 the set P(ϵ) contains infinitely many consecutive pairs n, n+1. This was proved recently by Hildebrand [4]. In the present paper we shall investigate quantitative forms of Hildebrand’s result.
Authors
Affiliations
1 Magdalen College, Oxford OX 1 4 AU, England, GB
1 Magdalen College, Oxford OX 1 4 AU, England, GB
Source
The Journal of the Indian Mathematical Society, Vol 52, No 1-2 (1987), Pagination: 39-49Abstract
For ϵ>0 we defineP(ϵ)={n∈N: n=pk, p prime, k≤n}.
Thus if ϵ is small, the elements of P(ϵ) are “almost prime”. (Note that this phrase is used here in a different sense from that of the weighted sieve, in which n=Pr is an almost-prime if n has at most r prime factors.) It was conjectured by Erdos [1] that for any ϵ>0 the set P(ϵ) contains infinitely many consecutive pairs n, n+1. This was proved recently by Hildebrand [4]. In the present paper we shall investigate quantitative forms of Hildebrand’s result.