scholarly journals ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES

2018 ◽  
Vol 107 (1) ◽  
pp. 133-144 ◽  
Author(s):  
JIE WU
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Prime Numbers ◽  
Asymptotic Density ◽  
Nonzero Integer ◽  
Shifted Primes ◽  
Image Position

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$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

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 Square-full numbers in short intervals

1991 ◽  
Vol 110 (1) ◽  
pp. 1-3 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Positive Integer ◽  
Riemann Hypothesis ◽  
Prime Factor ◽  
Short Interval ◽  
Short Intervals ◽  
Image Position

A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] thatBateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we havewhen and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

scholarly journals On a sequence of prime numbers

1968 ◽  
Vol 8 (3) ◽  
pp. 571-574 ◽  
Author(s):  
C. D. Cox ◽  
A. J. Van Der Poorten
Keyword(s):  
Prime Factor ◽  
Prime Numbers ◽  
Image Position

Euclid's scheme for proving the infinitude of the primes generates, amongst others, the following sequence defined by p1 = 2 and pn+1 is the highest prime factor of p1p2…pn+1.

scholarly journals On sparsely totient numbers

1991 ◽  
Vol 33 (3) ◽  
pp. 350-358
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Multiplicative Structure ◽  
Euler Totient Function ◽  
Image Position

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient ifHere φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) thatand infinitely often

The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

2020 ◽  
pp. 1-12
Author(s):  
Paul Pollack
Keyword(s):  
Positive Integer ◽  
Multiplicative Group ◽  
Asymptotic Density ◽  
Normal Order ◽  
Sylow Subgroups ◽  
Group Of Units ◽  
Euler’S Function ◽  
Image Position ◽  
Group Modulo

Abstract For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$ , $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$ .

scholarly journals ON THE LARGEST PRIME FACTOR OF THE MERSENNE NUMBERS

2009 ◽  
Vol 79 (3) ◽  
pp. 455-463 ◽  
Author(s):  
KEVIN FORD ◽  
FLORIAN LUCA ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Positive Integer ◽  
London Math ◽  
Prime Factor ◽  
Precise Form ◽  
Mersenne Numbers ◽  
Image Position ◽  
Lehmer Numbers

AbstractLetP(k) be the largest prime factor of the positive integerk. In this paper, we prove that the seriesis convergent for each constantα<1/2, which gives a more precise form of a result of C. L. Stewart [‘On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers’,Proc. London Math. Soc.35(3) (1977), 425–447].

The square-full numbers in an interval

1996 ◽  
Vol 119 (2) ◽  
pp. 201-208 ◽  
Author(s):  
M. N. Huxley ◽  
O. Trifonov
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Perfect Squares ◽  
Image Position

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

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