On shifted primes with large prime factors and their
      products

Luca, Florian; Menares, Ricardo; Pizarro-Madariaga, Amalia

doi:10.36045/bbms/1426856856

Cited by 14 publications

(14 citation statements)

References 7 publications

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“…In addition, it should be noted that we can obtain a result similar to (1.8) directly for the small values of β by a result of Fouvry [18,Théorème 2], a theorem of Bombieri-Vinogradov type on sifted integers (without small prime factor) with an exponent of distribution > 1 2 , or applying a weighted sieve combined with a theorem of Bombieri-Vinogradov type (see Fouvry [8, Théorème 2; 7, Corollaire 3]).…”

Section: Theorem 1 For X → ∞ and Ysupporting

confidence: 54%

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On the smooth values of shifted almost-primes

Lü

Wang

Chen

2019

Int. J. Number Theory

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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.

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Section: Theorem 1 For X → ∞ and Ysupporting

confidence: 54%

“…and we want to estimate T θ (x) with a good lower bound (see [18,3,6]). As in the very recent work of Liu, Wu and Xi [17], with the help of (1.6), for x → ∞, we have the following lower bound:…”

Section: Theorem 1 For X → ∞ and Ymentioning

confidence: 99%

On the smooth values of shifted almost-primes

Lü

Wang

Chen

2019

Int. J. Number Theory

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“…For example, we are interested in the greatest value of θ for which there is a positive proportion of primes p such that P (p − a) p θ (see [7,6,1,17,14]). For given θ ∈ (0, 1), we also considered the relative asymptotic density of such primes in P (see [12,3,5]). Motived by these questions, Liu, Wu & Xi [11] studied the distribution of primes in arithmetic progressions with friable indices, i.e., {a + mq} m friable (recall that a positive integer m is friable if its all prime factors are small), and established analogues of classical Siegel-Walfisz theorem, Bombieri-Vinogradov theorem and Brun-Titchmarsh theorem.…”

Section: Introductionmentioning

confidence: 99%

On Values Taken by the Largest Prime Factor of Shifted Primes

2018

J. Aust. Math. Soc.

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Denote by $\mathbb{P}$ the set of all prime numbers and by $P(n)$ the largest prime factor of positive integer $n\geq 1$ with the convention $P(1)=1$. In this paper, we prove that, for each $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant $c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer $a\in \mathbb{Z}^{\ast }$, the set $$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$ has relative asymptotic density one in $\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’, J. Aust. Math. Soc.82 (2015), 133–147], Theorem 1.1, which requires $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$ in place of $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

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“…As usual, we use P + (n) to denote the largest prime factor of n with the convention P + (1) = 1. Recently Luca, Menares and Pizarro-Madariaga [9] considered the following counting function (1.1) T θ (x) := p x : P + (p − 1) p θ and proved (see [9, Theorem 1])…”

Section: Introductionmentioning

confidence: 99%

“…where π(x) denotes the number of primes p x, the implied constant depends on θ only and [3] on the modularity of reducible mod Galois representations, Luca, Menares and Pizarro-Madariaga, in the same paper [9], also studied the following counting function…”

Section: Introductionmentioning

confidence: 99%