scholarly journals A probabilistic solution for the Syracuse conjecture

2022 ◽  
Author(s):  
El ghazi Imad
Keyword(s):  
Positive Integer ◽  
Arbitrary Positive Integer ◽  
Probabilistic Solution

Abstract We prove the veracity of the Syracuse conjecture by establishingthat starting from an arbitrary positive integer diffrent of 1 and 4, theSyracuse process will never comeback to any positive integer reachedbefore and then we conclude by using a probabilistic approach.Classification : MSC: 11A25

scholarly journals A probabilistic solution for the Syracuse conjecture

2021 ◽  
Author(s):  
El ghazi Imad
Keyword(s):  
Random Walk ◽  
Positive Integer ◽  
Arbitrary Positive Integer ◽  
Probabilistic Solution

Abstract We prove the veracity of the Syracuse conjecture by establishing that starting from an arbitrary positive integer, the Syracuse process will never reach any integer reached before and then we conclude by using a probabilistic (a random walk) approach.Classification: MSC: 11A25

scholarly journals Anti-integral extensions $ {R[{\alpha}]/R$

2004 ◽  
Vol 35 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Kiyoshi Baba ◽  
Ken-Ichi Yoshida
Keyword(s):  
Positive Integer ◽  
Integral Domain ◽  
Arbitrary Positive Integer ◽  
Integral Element ◽  
The Ideal

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

scholarly journals A further result on the complex oscillation theory of periodic second order linear differential equations

1990 ◽  
Vol 33 (1) ◽  
pp. 143-158 ◽  
Author(s):  
Shian Gao
Keyword(s):  
Entire Function ◽  
Positive Integer ◽  
Oscillation Theory ◽  
Transcendental Entire Function ◽  
Arbitrary Positive Integer ◽  
Order Of Growth ◽  
Exponent Of Convergence ◽  
Complex Oscillation ◽  
Zero Sequence ◽  
Linear Differential

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

scholarly journals A Sum Connected with Quadratic Residues

1956 ◽  
Vol 10 ◽  
pp. 1-7
Author(s):  
L. Carlitz
Keyword(s):  
Positive Integer ◽  
Legendre Symbol ◽  
Arbitrary Positive Integer ◽  
Arbitrary Integer ◽  
Quadratic Residues

Let p be a prime > 2 and m an arbitrary positive integer; definewhere (r/p) is the Legendre symbol. We consider the problem of finding the highest power of p dividing Sm. A little more generally, if we putwhere a is an arbitrary integer, we seek the highest power of p dividing Sm(a). Clearly Sm = Sm(0), and Sm(a) = Sm(b) when a ≡ b (mod p).

scholarly journals On rings all of whose factor rings are integral domains

1993 ◽  
Vol 55 (3) ◽  
pp. 325-333
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Positive Integer ◽  
Zero Divisor ◽  
Arbitrary Positive Integer ◽  
Integral Domains ◽  
Factor Ring ◽  
Zero Divisors ◽  
Proper Quotient

AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

Finite order corks

2017 ◽  
Vol 28 (06) ◽  
pp. 1750034 ◽  
Author(s):  
Motoo Tange
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Stein Manifold ◽  
Contact Structures ◽  
Arbitrary Positive Integer ◽  
Branched Cover

We show that for any positive integer [Formula: see text], there exist pairs of compact, contractible, Stein 4-manifolds and order [Formula: see text] self-diffeomorphisms of the boundaries that do not extend to the full manifolds. Each boundary of the Stein 4-manifolds is a cyclic branched cover along a slice knot embedded in the boundary of a contractible 4-manifold. Each pair is called a finite order cork, we give a method producing examples of many finite order corks, which are possibly not a Stein manifold. The example of the Stein cork gives a diffeomorphism generating [Formula: see text] homotopic but non-isotopic Stein fillable contact structures for an arbitrary positive integer [Formula: see text].

On the digits of the multiples of an irrational p-adic number

1974 ◽  
Vol 76 (2) ◽  
pp. 417-422
Author(s):  
Kurt Mahler
Keyword(s):  
Positive Integer ◽  
Rational Integer ◽  
Arbitrary Positive Integer ◽  
Image Position

Let α be an irrational p-adic number, r an arbitrary positive integer. Our aim is to prove that there exists a rational integer X satisfyingsuch that every possible sequence of r digits 0, 1, …, p – 1 occurs infinitely often in the canonical p-adic series for Xα. It is clear that it suffices to prove this result for p-adic integers.

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