scholarly journals Introduction to Diophantine Approximation

2015 ◽  
Vol 23 (2) ◽  
pp. 101-106
Author(s):  
Yasushige Watase
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Continued Fractions ◽  
Irrational Number ◽  
Integer Solution ◽  
Infinitely Many Solutions

Abstract In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

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 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].

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

scholarly journals Introduction to Diophantine Approximation. Part II

2017 ◽  
Vol 25 (4) ◽  
pp. 283-288
Author(s):  
Yasushige Watase
Keyword(s):  
Diophantine Approximation ◽  
Irrational Number ◽  
Basic Result ◽  
Formal Proof ◽  
Integer Solution ◽  
Linear Forms ◽  
Diophantine Approximations ◽  
Integer Solutions ◽  
System 1 ◽  
Hurwitz Theorem

SummaryIn the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2− a2b1| ≠ 0, has at least one integer solution.

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].

scholarly journals Some inequalities for diophantine approximation by continued fractions

1990 ◽  
Vol 41 (2) ◽  
pp. 249-253
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
General Method

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,a2,…,an,…], let pn/qn be its nth convergent and let θn = qn|qnξ − pn|. In this paper a general method is introduced to deduce a series of inequalities involving the triple (θn−1, θn, θn+1).

scholarly journals A problem of Hooley in Diophantine approximation

1996 ◽  
Vol 38 (3) ◽  
pp. 299-308
Author(s):  
Glyn Harman
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Image Position

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequalityHere

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.

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