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.

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].

One Diophantine inequality with unlike powers of prime variables

2016 ◽  
Vol 13 (06) ◽  
pp. 1531-1545 ◽  
Author(s):  
Quanwu Mu
Keyword(s):  
Recent Result ◽  
Real Number ◽  
Infinitely Many Solutions ◽  
Diophantine Inequality ◽  
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 of the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text], where [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text]. This gives an improvement of the recent result.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals ON THE SPECTRA OF PISOT NUMBERS

2011 ◽  
Vol 54 (1) ◽  
pp. 127-132 ◽  
Author(s):  
TOUFIK ZAIMI
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Pisot Number ◽  
Real Numbers ◽  
Pisot Numbers

AbstractLet θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all$\alpha \in \{ \theta ^{n}\mid n\in \mathbb{N}\} \cup \{ \sum\nolimits_{n=0}^{N}\theta ^{n}\mid \mathit{\}N\in \mathbb{N}\}$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such that$\Vert \lambda \alpha \Vert <\frac{1}{% 3}$for all$\alpha \in \{ \sum\nolimits_{n=0}^{N}a_{n}\theta ^{n}\mid$an ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

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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