Research

12 pages, 251 KiB

Open AccessArticle

On the Number of Witnesses in the Miller–Rabin Primality Test

by Shamil Talgatovich Ishmukhametov, Bulat Gazinurovich Mubarakov and Ramilya Gakilevna Rubtsova

Symmetry 2020, 12(6), 890; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12060890 - 01 Jun 2020

Cited by 7 | Viewed by 3068

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let [...] Read more.

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let

W (n)

denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer

a < n

is a primality witness for n. For composite n, the power of

W (n)

is less than or equal to

φ (n) / 4

where

φ (n)

is Euler’s Totient function. We derive new exact formulas for the power of

W (n)

depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

8 pages, 209 KiB

Open AccessArticle

On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

by Atsushi Yamagami and Kazuki Taniguchi

Symmetry 2020, 12(2), 288; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12020288 - 16 Feb 2020

Cited by 1 | Viewed by 1799

Abstract

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer

k \geq 2

. Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the [...] Read more.

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer

k \geq 2

. Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the

p^{e}

-Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form

\frac{p^{e m} - 1}{p^{e} - 1}

with some integer

m \geq 1

. This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence

{(x + 1)}^{p^{e}} \equiv {(x^{p} + 1)}^{p^{e - 1}} (\mod p^{e})

of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

26 pages, 1071 KiB

Open AccessArticle

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

by Torsten Asselmeyer-Maluga

Symmetry 2019, 11(10), 1298; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11101298 - 15 Oct 2019

Cited by 11 | Viewed by 4212

Abstract

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement [...] Read more.

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement

C (K) = S^{3} \ (K \times D^{2})

of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of

S^{3}

branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk

D^{3}

branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model). Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

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5 pages, 223 KiB

Open AccessArticle

Algebraic Numbers as Product of Powers of Transcendental Numbers

by Pavel Trojovský

Symmetry 2019, 11(7), 887; https://doi.org/10.3390/sym11070887 - 08 Jul 2019

Cited by 3 | Viewed by 2235

Abstract

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in

C [x]

, and the problem of finding zeros in

Q [x]

leads to the definition of algebraic and transcendental numbers. Recently, [...] Read more.

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in

C [x]

, and the problem of finding zeros in

Q [x]

leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form

P {(T)}^{Q (T)}

. In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form

P_{1} {(T)}^{Q_{1} (T)} \dots P_{n} {(T)}^{Q_{n} (T)}

for some transcendental number T, where

P_{1}, \dots, P_{n}, Q_{1}, \dots, Q_{n}

are prescribed, non-constant polynomials in

Q [x]

(under weak conditions). More generally, our result generalizes results on the arithmetic nature of

z^{w}

when z and w are transcendental. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

31 pages, 330 KiB

Open AccessArticle

On Some Formulas for Kaprekar Constants

by Atsushi Yamagami and Yūki Matsui

Symmetry 2019, 11(7), 885; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11070885 - 05 Jul 2019

Cited by 2 | Viewed by 3020

Abstract

Let

b \geq 2

and

n \geq 2

be integers. For a b-adic n-digit integer x, let A (resp. B) be the b-adic n-digit integer obtained by rearranging the numbers of all digits of x in descending [...] Read more.

Let

b \geq 2

and

n \geq 2

be integers. For a b-adic n-digit integer x, let A (resp. B) be the b-adic n-digit integer obtained by rearranging the numbers of all digits of x in descending (resp. ascending) order. Then, we define the Kaprekar transformation

T_{(b, n)} (x) : = A - B

. If

T_{(b, n)} (x) = x

, then x is called a b-adic n-digit Kaprekar constant. Moreover, we say that a b-adic n-digit Kaprekar constant x is regular when the numbers of all digits of x are distinct. In this article, we obtain some formulas for regular and non-regular Kaprekar constants, respectively. As an application of these formulas, we then see that for any integer

b \geq 2

, the number of b-adic odd-digit regular Kaprekar constants is greater than or equal to the number of all non-trivial divisors of b. Kaprekar constants have the symmetric property that they are fixed points for recursive number theoretical functions

T_{(b, n)}

. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

33 pages, 419 KiB

Open AccessArticle

Asymptotic Semicircular Laws Induced by p-Adic Number Fields ℚ p and C^*-Algebras over Primes p

by Ilwoo Cho

Symmetry 2019, 11(6), 819; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11060819 - 20 Jun 2019

Viewed by 1867

Abstract

In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed

C^{*}

-probability spaces, and p-adic number fields

{Q_{p}}_{p \in P}

, as p→ ∞ in the set

P

of all primes. [...] Read more.

In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed

C^{*}

-probability spaces, and p-adic number fields

{Q_{p}}_{p \in P}

, as p→ ∞ in the set

P

of all primes. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

22 pages, 454 KiB

Open AccessArticle

An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure

by Bruno Aiazzi, Stefano Baronti, Leonardo Santurri and Massimo Selva

Symmetry 2019, 11(6), 775; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11060775 - 10 Jun 2019

Cited by 1 | Viewed by 3254

Abstract

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the [...] Read more.

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the case of twin primes is formulated. The proposed investigation, which is named Limited INtervals into PEriodical Sequences (LINPES) relies on a set of binary periodical sequences that are evaluated in limited intervals of the prime characteristic function. These sequences are built by considering the ensemble of deleted (that is, 0) and undeleted (that is, 1) integers in a modified version of the Sieve procedure, in such a way a symmetric succession of runs of zeroes is found in correspondence of the gaps between the undeleted integers in each period. Such a formulation is able to estimate the prime number function in an equivalent way to the logarithmic integral function Li(x). The present analysis is then extended to the twin primes, by taking into account only the runs whose size is two. In this case, the proposed procedure gives an estimation of the twin prime function that is equivalent to the one of the logarithmic integral function

{Li}_{2} (x)

. As a consequence, a possibility is investigated in order to count the twin primes in the same intervals found for the primes. Being that the bounds of these intervals are given by squares of primes, if such an inference were actually proved, then the twin primes could be estimated up to infinity, by strengthening the conjecture of their never-ending. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

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37 pages, 1934 KiB

Open AccessArticle

The Riemann Zeros as Spectrum and the Riemann Hypothesis

by Germán Sierra

Symmetry 2019, 11(4), 494; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11040494 - 04 Apr 2019

Cited by 15 | Viewed by 7061

Abstract

We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a [...] Read more.

We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

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15 pages, 1556 KiB

Open AccessArticle

Universal Quantum Computing and Three-Manifolds

by Michel Planat, Raymond Aschheim, Marcelo M. Amaral and Klee Irwin

Symmetry 2018, 10(12), 773; https://0-doi-org.brum.beds.ac.uk/10.3390/sym10120773 - 19 Dec 2018

Cited by 12 | Viewed by 4902

Abstract

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere

S^{3}

. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the [...] Read more.

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere

S^{3}

. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold

M^{3}

. More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group

P S L (2, Z)

correspond to d-fold

M^{3}

- coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and

M^{3}

’s obtained from Dehn fillings are explored. Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

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