scholarly journals On Diophantine approximation by unlike powers of primes

2019 ◽  
Vol 17 (1) ◽  
pp. 544-555
Author(s):  
Wenxu Ge ◽  
Weiping Li ◽  
Tianze Wang
Keyword(s):  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Abstract Suppose that λ1, λ2, λ3, λ4, λ5 are nonzero real numbers, not all of the same sign, λ1/λ2 is irrational, λ2/λ4 and λ3/λ5 are rational. Let η real, and ε > 0. Then there are infinitely many solutions in primes pj to the inequality $\begin{array}{} \displaystyle |\lambda_1p_1+\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^4+\lambda_5p_5^5+\eta| \lt (\max{p_j^j})^{-1/32+\varepsilon} \end{array}$. This improves an earlier result under extra conditions of λj.

scholarly journals Diophantine approximation with mixed powers of primes

2018 ◽  
Vol 14 (07) ◽  
pp. 1903-1918
Author(s):  
Wenxu Ge ◽  
Huake Liu
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Let [Formula: see text] be an integer with [Formula: see text], and [Formula: see text] be any real number. Suppose that [Formula: see text] are nonzero real numbers, not all the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in primes [Formula: see text], where [Formula: see text], and [Formula: see text] for [Formula: see text]. This generalizes earlier results. As application, we get that the integer parts of [Formula: see text] are prime infinitely often for primes [Formula: see text].

Some metrical theorems in diophantine approximation

1951 ◽  
Vol 47 (1) ◽  
pp. 18-21 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Solutions ◽  
The Difference ◽  
Image Position

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequalityhas infinitely many integer solutions q > 0. In particular, if α is any real number, the inequalityhas infinitely many solutions.

scholarly journals Diophantine approximation with one prime, two squares of primes and one kth power of a prime

2021 ◽  
Vol 19 (1) ◽  
pp. 373-387
Author(s):  
Alessandro Gambini
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Abstract Let 1 < k < 14 / 5 1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5 , λ 1 , λ 2 , λ 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and λ 4 {\lambda }_{4} be non-zero real numbers, not all of the same sign such that λ 1 / λ 2 {\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational and let ω \omega be a real number. We prove that the inequality ∣ λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 k − ω ∣ ≤ ( max ( p 1 , p 2 2 , p 3 2 , p 4 k ) ) − ψ ( k ) + ε | {\lambda }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega | \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon } has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 {p}_{1},{p}_{2},{p}_{3},{p}_{4} for any ε > 0 \varepsilon \gt 0 , where ψ ( k ) = min 1 14 , 14 − 5 k 28 k \psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right) .

Diophantine approximation by unlike powers of primes

2017 ◽  
Vol 13 (09) ◽  
pp. 2445-2452 ◽  
Author(s):  
Zhixin Liu
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Let [Formula: see text] be nonzero real numbers not all of the same sign, satisfying that [Formula: see text] is irrational, and [Formula: see text] be a real number. In this paper, we prove that for any [Formula: see text] [Formula: see text] has infinitely many solutions in prime variables [Formula: see text].

A note on Diophantine approximation by unlike powers of primes

2018 ◽  
Vol 14 (06) ◽  
pp. 1651-1668 ◽  
Author(s):  
Quanwu Mu ◽  
Yunyun Qu
Keyword(s):  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

It is proved that if [Formula: see text] are nonzero real numbers, not all of the same sign and [Formula: see text] is irrational, then for given real numbers [Formula: see text] and [Formula: see text], [Formula: see text], the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text]. This result constitutes an improvement upon that of Liu for the range [Formula: see text].

Approximation by a sum of polynomials of different degrees involving primes

1979 ◽  
Vol 27 (4) ◽  
pp. 454-466
Author(s):  
Ming-Chit Liu
Keyword(s):  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Coefficients ◽  
Image Position

AbstractLet λj (1 ≦ j ≦ 4) be any nonzero real numbers which are not all of the same sign and not all in rational ratio and let pj be polynomials of degree one or two with integer coefficients and positive leading coefficients. The author proves that if exactly two pj are of degree two then for any real n there are infinitely many solutions in primes pj of the inequality . where 0 <β < (√(21)–1)∖5760.

scholarly journals Uniform Diophantine approximation related to -ary and -expansions

2014 ◽  
Vol 36 (1) ◽  
pp. 1-22 ◽  
Author(s):  
YANN BUGEAUD ◽  
LINGMIN LIAO
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Large Integer ◽  
Real Numbers

Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$. We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$, where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$-transformation.

scholarly journals A Diophantine Problem with Unlike Powers of Primes

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Quanwu Mu ◽  
Liyan Xi
Keyword(s):  
Real Number ◽  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Diophantine Problem

Let k be an integer with 4 ≤ k ≤ 6 and η be any real number. Suppose that λ 1 , λ 2 , … , λ 5 are nonzero real numbers, not all of them have the same sign, and λ 1 / λ 2 is irrational. It is proved that the inequality λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 3 + λ 4 p 4 4 + λ 5 p 5 k + η < max 1 ≤ j ≤ 5 p j − σ k has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 ,  and  p 5 , where 0 < σ 4 < 1 / 36 , 0 < σ 5 < 4 / 189 , and 0 < σ 6 < 1 / 54 . This gives an improvement of the recent results.

Diophantine Approximation and Horocyclic Groups

1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Modular Group ◽  
Positive Real Number ◽  
Irrational Number ◽  
Infinitely Many Solutions ◽  
Positive Real ◽  
Closed Set ◽  
Coprime Integers ◽  
Image Position

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

scholarly journals On the critical determinants of certain star bodies

2017 ◽  
Vol 25 (1) ◽  
pp. 5-11 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Calculus Of Variations ◽  
Diophantine Approximation ◽  
Star Body ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Critical Determinant ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies ◽  
Classic Paper

Abstract In a classic paper [14], W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body|x1|(|x1|3 + |x2|3 + |x3|3 ≤ 1.In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body|x1|(|x1|3 + |x22 + x32)3/2≤ 1.

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