Entiers sans grand ni petit facteur premier III

Saïas, Éric

doi:10.4064/aa-71-4-351-379

Cited by 7 publications

(4 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We note that by [4, Theorem 1] of Friedlander, we can obtain an asymptotic formula for Ψ(x; y, z) in certain regions. We also mention that Saias [15,16,17] has extensively studied this quantity. In particular, our Ψ(x k ; y, z) is θ(x k , z, y) or Θ(x k , z, y) in the notation of Friedlander and Saias respectively.…”

Section: Resultsmentioning

confidence: 99%

Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

Yau

2019

Acta Arith.

View full text Add to dashboard Cite

For any ε > 0, we obtain an asymptotic formula for the number of solutions n ≤ x to αn + β < x −1/4+ε where n is [y, z]-smooth for infinitely many real numbers x. In addition, we also establish an asymptotic formula with an additional square-free condition on n. Moreover, if α is quadratic irrational then the asymptotic formulas hold for all sufficiently large x.Our tools come from the Harman sieve which we adapt suitably to sieve for [y, z]-smooth numbers. The arithmetic information comes from estimates for exponential sums.

show abstract

Section: Resultsmentioning

confidence: 99%

Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

Yau

2019

Acta Arith.

View full text Add to dashboard Cite

show abstract

“…From (1.7) in Theorem 2 in Saias (1993) and from Theorem 1 and (iv), (v) of Lemma 4 in Saias (1992) (cf. Theorem 6 of Friedlander 1976) we have 1…”

Section: A Combinatorial Beginningmentioning

confidence: 99%

A hyperelliptic smoothness test. I

Lenstra

Pila²,

Pomerance

1993

Phil. Trans. R. Soc. Lond. A

View full text Add to dashboard Cite

This series of papers is concerned with a probabilistic algorithm for finding small prime factors of an integer. While the algorithm is not practical, it yields an improvement over previous complexity results. The algorithm uses the jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. In this first paper in the series a new density theorem is presented for smooth numbers in short intervals. It is a key ingredient of the analysis of the algorithm.

show abstract

“…where P − (n) denotes the smallest prime factor of integer n with the convention P − (1) = ∞. For the case a = 0, we may refer to the work of Friedlander [11] and Saias [20][21][22]. Here, we only consider the case a = 0.…”

Section: Introductionmentioning

confidence: 99%

On the smooth values of shifted almost-primes

Lü

Wang

Chen

2019

Int. J. Number Theory

View full text Add to dashboard Cite

Denote by [Formula: see text] (respectively, [Formula: see text]) the largest (respectively, the smallest) prime factor of the integer [Formula: see text]. In this paper, we prove a lower bound of almost-primes [Formula: see text] with [Formula: see text] such that [Formula: see text] for [Formula: see text]. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor.

show abstract

Entiers sans grand ni petit facteur premier III

Cited by 7 publications

References 2 publications

Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

A hyperelliptic smoothness test. I

On the smooth values of shifted almost-primes

Contact Info

Product

Resources

About