scholarly journals PRODUCTS OF SHIFTED PRIMES SIMULTANEOUSLY TAKING PERFECT POWER VALUES

2012 ◽  
Vol 92 (2) ◽  
pp. 145-154 ◽  
Author(s):  
TRISTAN FREIBERG
Keyword(s):  
Lower Bound ◽  
Prime Factors ◽  
Squarefree Integers ◽  
Shifted Primes

AbstractLet r be an integer greater than 1, and let A be a finite, nonempty set of nonzero integers. We obtain a lower bound for the number of positive squarefree integers n, up to x, for which the products ∏ p∣n(p+a) (over primes p) are perfect rth powers for all the integers a in A. Also, in the cases where A={−1} and A={+1}, we will obtain a lower bound for the number of such n with exactly r distinct prime factors.

scholarly journals On shifted primes with large prime factors and their products

2015 ◽  
Vol 22 (1) ◽  
pp. 39-47 ◽  
Author(s):  
Florian Luca ◽  
Ricardo Menares ◽  
Amalia Pizarro-Madariaga
Keyword(s):  
Prime Factors ◽  
Shifted Primes

scholarly journals Shifted primes without large prime factors

1998 ◽  
Vol 83 (4) ◽  
pp. 331-361 ◽  
Author(s):  
R. Baker ◽  
G. Harman
Keyword(s):  
Prime Factors ◽  
Shifted Primes

scholarly journals On the smooth values of shifted almost-primes

2019 ◽  
Vol 15 (01) ◽  
pp. 1-9
Author(s):  
Xiaodong Lü ◽  
Zhiwei Wang ◽  
Bin Chen
Keyword(s):  
Lower Bound ◽  
Prime Factor ◽  
Almost Primes ◽  
Prime Factors ◽  
Consecutive Integers

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.

Short intervals containing numbers without large prime factors

1991 ◽  
Vol 109 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Riemann Hypothesis ◽  
Short Intervals ◽  
Prime Factors ◽  
Image Position

We denote, as usual, the number of integers not exceeding x having no prime factors greater than y by Ψ(x, y). We also writeThe function Ψ(x, y) is of great interest in number theory and has been studied by many researchers (see [3], [5] and [6] for example). The function Ψ(x, z, y) has also received some attention (see [2], [4–6]). In this paper we shall try to obtain a positive lower bound for Ψ(x, z, y) with y as small as possible when z is about x½ in magnitude. We note that the approach in [5] and [6] allows y to be much smaller than is permissible here, but requires x/z to be smaller than any power of x in [6] (unless some conjecture like the Riemann Hypothesis is assumed), or needs in [5]. The following result was obtained by Balog[1].

scholarly journals On the density of shifted primes with large prime factors

2017 ◽  
Vol 61 (1) ◽  
pp. 83-94 ◽  
Author(s):  
Bin Feng ◽  
Jie Wu
Keyword(s):  
Prime Factors ◽  
Shifted Primes

scholarly journals On values taken by the largest prime factor of shifted primes

2007 ◽  
Vol 82 (1) ◽  
pp. 133-147 ◽  
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Prime Divisor ◽  
Prime Factor ◽  
Prime Numbers ◽  
Asymptotic Density ◽  
Shifted Primes

AbstractLet P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n > 1. We show that, for every real number , there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c(η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P(q - a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log logq)1+o(1) times before it terminates.

scholarly journals On a Multiplicative Partition Function

10.37236/1563 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Yifan Yang
Keyword(s):  
Partition Function ◽  
Dirichlet Series ◽  
Asymptotic Estimate ◽  
Infinite Product ◽  
Prime Factors ◽  
Squarefree Integers ◽  
Multiplicative Partition Function

Let $D(s)=\sum^\infty_{m=1}a_mm^{-s}$ be the Dirichlet series generated by the infinite product $\prod^\infty_{k=2}(1-k^{-s})$. The value of $a_m$ for squarefree integers $m$ with $n$ prime factors depends only on the number $n$, and we let $f(n)$ denote this value. We prove an asymptotic estimate for $f(n)$ which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma.

scholarly journals On values taken by the largest prime factor of shifted primes (II)

2019 ◽  
Vol 15 (05) ◽  
pp. 935-944 ◽  
Author(s):  
Bin Chen ◽  
Jie Wu
Keyword(s):  
Lower Bound ◽  
Asymptotic Formula ◽  
Relative Density ◽  
Positive Solution ◽  
Prime Factor ◽  
Positive Density ◽  
Unique Positive Solution ◽  
Shifted Primes ◽  
The Cost

Denote by [Formula: see text] the set of all primes and by [Formula: see text] the largest prime factor of integer [Formula: see text] with the convention [Formula: see text]. Let [Formula: see text] be the unique positive solution of the equation [Formula: see text] in [Formula: see text]. Very recently Wu proved that for [Formula: see text] there is a constant [Formula: see text] such that for each fixed non-zero integer [Formula: see text] the set [Formula: see text] has relative density 1 in [Formula: see text]. In this paper, we shall further extend the domain of [Formula: see text] at the cost of obtaining a lower bound in place of an asymptotic formula, by showing that for each [Formula: see text] the set [Formula: see text] has relative positive density in [Formula: see text].

Linear Equations with Small Prime and Almost Prime Solutions

2008 ◽  
Vol 51 (3) ◽  
pp. 399-405
Author(s):  
Xianmeng Meng
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Equations ◽  
Prime Factors ◽  
Almost All ◽  
Sharp Lower Bound ◽  
Almost Prime

AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.

Sign in / Sign up

Export Citation Format

Share Document