Irregularities in the Distribution of Prime Numbers in a Beatty Sequence

2019 ◽  
Vol 63 (4) ◽  
pp. 738-743
Author(s):  
Janyarak Tongsomporn ◽  
Jörn Steuding
Keyword(s):  
Finite Type ◽  
Irrational Number ◽  
Prime Numbers ◽  
Positive Real ◽  
Beatty Sequence

AbstractWe prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

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 Independence of -adic Galois representations over function fields

2013 ◽  
Vol 149 (7) ◽  
pp. 1091-1107 ◽  
Author(s):  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Galois Representations ◽  
Function Fields ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Prime Numbers ◽  
Positive Answer ◽  
Absolute Galois Group ◽  
The Family

AbstractLet$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

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