## On a combinatorial method for counting smooth numbers in sets of integers

Venue: | J. Number Theory |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Croot_ona,

author = {Ernie Croot},

title = {On a combinatorial method for counting smooth numbers in sets of integers},

journal = {J. Number Theory},

year = {},

pages = {237--253}

}

### OpenURL

### Abstract

In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this result is completely combinatorial and elementary. 1

### Citations

45 |
On the frequency of numbers containing prime factors of a certain relative magnitude, Ark
- Dickman
- 1930
(Show Context)
Citation Context ... of “shifted primes”. It is conjectured that Ψ(S, x θ ) ∼ π(x)Ψ(x, xθ ) x ∼ ρ(θ −1 )π(x), (2) 1where ρ(u) = lim x→∞ Ψ(x, x1/u ) . x This function ρ is called Dickman’s function, and it was proved in =-=[7]-=- that the limit exists. Unfortunately, proving (2) remains a difficult, open problem; however, in [9], J. B. Friedlander gave a beautiful proof that Ψ(S, x θ ) ≫ π(x) for θ > (2 √ e) −1 , and in [1], ... |

35 |
Shifted primes without large prime factors
- Baker, Harman
- 1998
(Show Context)
Citation Context ...n [7] that the limit exists. Unfortunately, proving (2) remains a difficult, open problem; however, in [9], J. B. Friedlander gave a beautiful proof that Ψ(S, x θ ) ≫ π(x) for θ > (2 √ e) −1 , and in =-=[1]-=-, R. Baker and G. Harman proved that for θ ≥ 0.2961, Ψ(S, x θ ) > x log α x , for some α > 1 and x > x0(a). There are several methods for attacking the general question of proving that (1) holds for a... |

17 |
Shifted primes without large prime factors, in Number Theory and Applications
- FRIEDLANDER
- 1989
(Show Context)
Citation Context ... ρ(u) = lim x→∞ Ψ(x, x1/u ) . x This function ρ is called Dickman’s function, and it was proved in [7] that the limit exists. Unfortunately, proving (2) remains a difficult, open problem; however, in =-=[9]-=-, J. B. Friedlander gave a beautiful proof that Ψ(S, x θ ) ≫ π(x) for θ > (2 √ e) −1 , and in [1], R. Baker and G. Harman proved that for θ ≥ 0.2961, Ψ(S, x θ ) > x log α x , for some α > 1 and x > x0... |

14 |
On differences and sums of integers
- Erdős, Sárközy
- 1977
(Show Context)
Citation Context ...iful papers by A. Balog and A. Sarkozy [2], [3], [4], and [5]; P. Erdős, H. Maier, and A. Sárkőzy [8]; A. Sárkőzy and C. L. Stewart [11], [12], [13], [14]; C. Pomerance, A. Sárkőzy, and C. L. Stewart =-=[10]-=-; and R. de la Bretèche [6]. The paper by de la Brèteche is more relevant to the main result of this section, and we give here one of his theorems: Theorem 3 Suppose that A and B are subsets of the in... |

5 |
On sums of Integers Having Small Prime Factors
- Balog, Sárkőzy
- 1984
(Show Context)
Citation Context ...denotes the largest prime factor of n. Using the large sieve and the circle method, these types of questions were given a thorough treatment in a series of beautiful papers by A. Balog and A. Sarkozy =-=[2]-=-, [3], [4], and [5]; P. Erdős, H. Maier, and A. Sárkőzy [8]; A. Sárkőzy and C. L. Stewart [11], [12], [13], [14]; C. Pomerance, A. Sárkőzy, and C. L. Stewart [10]; and R. de la Bretèche [6]. The paper... |

3 |
la Bretèche, ‘Sommes sans grand facteur premier
- de
- 1999
(Show Context)
Citation Context ...A. Sarkozy [2], [3], [4], and [5]; P. Erdős, H. Maier, and A. Sárkőzy [8]; A. Sárkőzy and C. L. Stewart [11], [12], [13], [14]; C. Pomerance, A. Sárkőzy, and C. L. Stewart [10]; and R. de la Bretèche =-=[6]-=-. The paper by de la Brèteche is more relevant to the main result of this section, and we give here one of his theorems: Theorem 3 Suppose that A and B are subsets of the integers in {1, 2, ..., x}. F... |

2 |
On the Distribution of the Number of Prime Factors of Sums a + b Trans
- Erdős, Maier, et al.
- 1987
(Show Context)
Citation Context ...e and the circle method, these types of questions were given a thorough treatment in a series of beautiful papers by A. Balog and A. Sarkozy [2], [3], [4], and [5]; P. Erdős, H. Maier, and A. Sárkőzy =-=[8]-=-; A. Sárkőzy and C. L. Stewart [11], [12], [13], [14]; C. Pomerance, A. Sárkőzy, and C. L. Stewart [10]; and R. de la Bretèche [6]. The paper by de la Brèteche is more relevant to the main result of t... |