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 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 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 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

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

PROPRIÉTÉS LOCALES DES CHIFFRES DES NOMBRES PREMIERS

2017 ◽  
Vol 18 (1) ◽  
pp. 189-224
Author(s):  
Bruno Martin ◽  
Christian Mauduit ◽  
Joël Rivat
Keyword(s):  
Asymptotic Formula ◽  
Additive Function ◽  
Prime Numbers ◽  
Exponential Sum ◽  
Image Position

Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum $$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$ where the summation runs over prime numbers $p$ and where $\unicode[STIX]{x1D6FD}\in \mathbb{R}$, $k\in \mathbb{Z}$, and $g:\mathbb{N}\rightarrow \mathbb{Z}$ is a strongly $b$-additive function such that $\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$.

Perfect powers in values of certain polynomials at integer points

1986 ◽  
Vol 99 (2) ◽  
pp. 195-207 ◽  
Author(s):  
T. N. Shorey
Keyword(s):  
Prime Factor ◽  
Integer Points ◽  
Greatest Prime Factor ◽  
Image Position

1. For an integer v > 1, we define P(v) to be the greatest prime factor of v and we write P(1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d1, …, dt with t ≥ 2 be distinct integers in the interval [1, k]. For integers l ≥ 2, y > 0 and b > 0 with P(b) ≤ k, we consider the equationPutso that ½ < vt ≤ ¾. If α > 1 and kα < m ≤ kl, then equation (1) implies thatfor 1 ≤ i ≤ t and hence

Note on the Calculation of Logarithms to a large number of places by means of Factor Tables

1880 ◽  
Vol 22 (5) ◽  
pp. 346-352
Author(s):  
J. W. L. Glaisher
Keyword(s):  
Convenient Method ◽  
Prime Factor ◽  
Prime Numbers ◽  
Simple Addition ◽  
Greatest Prime Factor ◽  
Valuable Complement ◽  
Special Notice

There are several cases in which the factors of a number may be of use in ordinary mathematical processes; as for example, in the calculation of logarithms. This, although not the raison d’être of such a table, is so important an application as to deserve special notice, for by its means the number of numbers whose logarithms are known is greatly extended. For example, Abraham Sharp’s table contains 61-decimal Briggian logarithms of primes to 1,100, so that the logarithms of all numbers whose greatest prime factor does not exceed this number, may be obtained by simple addition. Similarly, Wolfram’s table gives 48-decimal hyperbolic logarithms of primes up to 10,009. Thus a factor table forms a very valuable complement to many-place logarithmic tables, the range of which must necessarily be comparatively limited ; and even in the case of large prime numbers, or numbers exceeding the limits of the factor table, it affords a very convenient method of calculating logarithms.

scholarly journals The quadratic reciprocity law and the elementary theta function

1985 ◽  
Vol 27 ◽  
pp. 19-30 ◽  
Author(s):  
Martin Eichler
Keyword(s):  
Modular Function ◽  
Number Fields ◽  
Prime Numbers ◽  
Legendre Symbol ◽  
Algebraic Number Fields ◽  
Quadratic Reciprocity ◽  
Reciprocity Law ◽  
Modular Subgroup ◽  
Image Position ◽  
Totally Real

This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b ℇ ℤ and a substitution of the modular subgroup Γ0(4). Then, if γ>0 and b≡1 mod 2,withandAccording to (1), the Legendre symbol behaves somewhat like a modular function ﹙apart from the known behaviour under and ﹚. (1) follows (see below) from the functional equationwithprovided thatHere we used and always will use the abbreviationand ℇδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ>0 (see also Shimura [5]).

scholarly journals A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

2017 ◽  
Vol 39 (4) ◽  
pp. 1042-1070 ◽  
Author(s):  
JOEL MOREIRA ◽  
FLORIAN KARL RICHTER
Keyword(s):  
Spectral Properties ◽  
Correspondence Principle ◽  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Ergodic Theorems ◽  
Ergodic Averages ◽  
Natural Conditions ◽  
Beatty Sequences ◽  
Image Position

We investigate how spectral properties of a measure-preserving system$(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$are reflected in the multiple ergodic averages arising from that system. For certain sequences$a:\mathbb{N}\rightarrow \mathbb{N}$, we provide natural conditions on the spectrum$\unicode[STIX]{x1D70E}(T)$such that, for all$f_{1},\ldots ,f_{k}\in L^{\infty }$,$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{ja(n)}f_{j}=\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{jn}f_{j}\end{eqnarray}$$in$L^{2}$-norm. In particular, our results apply to infinite arithmetic progressions,$a(n)=qn+r$, Beatty sequences,$a(n)=\lfloor \unicode[STIX]{x1D703}n+\unicode[STIX]{x1D6FE}\rfloor$, the sequence of squarefree numbers,$a(n)=q_{n}$, and the sequence of prime numbers,$a(n)=p_{n}$. We also obtain a new refinement of Szemerédi’s theorem via Furstenberg’s correspondence principle.

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.

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