Number Theory Research Articles

String art with repeating decimals

Rational numbers are typically represented as decimals or fractions. This paper explores the artistic potential of these representations, focusing on converting decimal expansions of rational numbers into visual forms via string art. By dividing a circle into 10 equal parts numbered from zero to nine, repeating decimals can be transformed into visual paths defined by the sequence of digits. This approach was used to create a pair of artworks and several interactive GeoGebra animations. The visual appeal of the designs is complemented by a mathematical explanation using number theory results.

All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2)

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form , where and are positive integers. Without loss of generality, we may assume that . This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation , by using elementary methods of number theory and computer calculations. In the process, we can see that . Then, we will consider separately the value of a positive integer in nine cases. The first case is impossible. For the second and third cases, we will separate to consider the value of . For the remaining cases, we will separate to consider the value of . The research results showed that all positive integer solutions of the Diophantine equation are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation , where is a positive integer with .

PHILOSOPHICAL VALUES IN NUMBER THEORY OF IKHWĀN AL-ṢAFĀ'

In mathematics, especially Arithmetic, there is a theory of numbers that can represent philosophical values. Mathematics is often presented with materialist values without deep meaning. One group of classical philosophers with mathematical thoughts, especially number theory, which explains the number material itself and relates it to meaningful philosophical values, is Ikhwān al-Ṣafā'. This research aims to reveal the philosophical values contained in the thought of the number. Using a qualitative research method based on a literature study approach, the researcher specifically examines the phenomenal book Rasā'il Ikhwān al-Ṣafā' as a primary source as well as other relevant references as secondary sources. The study found that in the number theory of Ikhwān al-Ṣafā', there are philosophical values such as monotheism described by the characteristics of the number one and the value of creation described by the sequence of the number ladder. This research initiates a discourse on the philosophy of numbers that previously has never been studied for it will serve as a pioneer for further research.

Elementary Number Theory Problems. Part XIII

Summary This paper formalizes problems 41, 92, 121–123, 172, 182, 183, 191, 192 and 192a from “250 Problems in Elementary Number Theory” by Wacław Sierpiński [8].

Elementary Number Theory Problems. Part XIV – Diophantine Equations

Summary This paper continues the formalization of chosen problems defined in the book “250 Problems in Elementary Number Theory” by Wacław Sierpiński.

The sum of four squares: An exploration of Lagrange's theorem and its legacy in number theory

Lagranges Four-square Theorem is a fundamental principle in number theory, which states that every positive integer can be expressed as the sum of four squares. The theorem was first conjectured by the Greek mathematician Diophantus of Alexandria in the 3rd century CE. It was later proved by Pierre de Fermat in the 17th century, and the first published proof was attributed to Joseph-Louis Lagrange in 1770. This paper presents a comprehensive account of the four-square theorem in number theory, which focuses on finding integer solutions to polynomial equations. The theorem has significantly advanced the study of Diophantine equations. It traces Lagranges Four-square Theorem from its conjectural origins to its emergence as a cornerstone of contemporary mathematical research. This paper reviews the proof of the theorem and its implications, as well as its connection to modern research and applications, highlighting its timeless relevance in mathematics. In addition, the paper reaffirms the extensive influence of the theorem on the advancement of Diophantine equations and its ongoing significance in elucidating the enigmas of number theory. This enhances our comprehension of the theorems position in the wider story of mathematical progress, confirming its significance in both historical and contemporary contexts.

The sum of four squares: An exploration of Lagrange's theorem and its legacy in number theory

Lagranges Four-square Theorem is a fundamental principle in number theory, which states that every positive integer can be expressed as the sum of four squares. The theorem was first conjectured by the Greek mathematician Diophantus of Alexandria in the 3rd century CE. It was later proved by Pierre de Fermat in the 17th century, and the first published proof was attributed to Joseph-Louis Lagrange in 1770. This paper presents a comprehensive account of the four-square theorem in number theory, which focuses on finding integer solutions to polynomial equations. The theorem has significantly advanced the study of Diophantine equations. It traces Lagranges Four-square Theorem from its conjectural origins to its emergence as a cornerstone of contemporary mathematical research. This paper reviews the proof of the theorem and its implications, as well as its connection to modern research and applications, highlighting its timeless relevance in mathematics. In addition, the paper reaffirms the extensive influence of the theorem on the advancement of Diophantine equations and its ongoing significance in elucidating the enigmas of number theory. This enhances our comprehension of the theorems position in the wider story of mathematical progress, confirming its significance in both historical and contemporary contexts.

Applications of Fourier transforms in engineering

The Fourier transform was proposed by Fourier in 1807. Fourier transform is a method of analyzing signals, which can analyze the components of the signal or synthesize the signal using these components. Many waveforms can be used as signal components, such as sine waves, square waves, sawtooth waves, etc. The Fourier transform uses sine waves as components of the signal. Due to its excellent properties, the Fourier transform has a wide range of applications in physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, optics, and other fields. This paper focuses on the study of fractional-order Fourier transform in the engineering field, for the equipment of the tiny fault diagnosis method, and according to some existing diagnostic methods, put forward the idea of diagnostic method enhancement.

Semi-Regular Continued Fractions with Fast-Growing Partial Quotients

In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field.

On the exponential Diophantine equation $x^2+p^mq^n=2y^p$

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.

Mini-Workshop: Felix Klein’s Foreign Students: Opening Up the Way for Transnational Mathematics

Extending existing analyses of the topic, the workshop aimed to investigate the influence of Felix Klein on the development of mathematics (especially number theory, algebra, geometry, analysis, applications of mathematics in scientific and technical fields as well as in mathematics education) in countries other than Germany. The goal of the workshop was to take a look at mathematicians of foreign origin who studied with Klein that have received little attention so far (including Czech, Greek, Hungarian, Japanese, Polish, Russian, and Ukrainian mathematicians) and uncover how Klein guided them through his lectures and seminars. The protocols of the lectures held in Klein’s seminars (from 1872 to 1912 in Göttingen, Erlangen, and Leipzig), which are a unique and so far largely unexplored source, were the basis for the workshop.

Quantum temporal logic and reachability problems of matrix semigroups

We study the reachability problems of a quantum finite automaton. More precisely, we introduce quantum temporal logic (QTL) that specifies the time-dependent behavior of quantum finite automaton by presenting the time dependence of events temporal operators ◊ (eventually) and □ (always) and employing the projections on subspaces as atomic propositions. The satisfiability of QTL formulae corresponds to various reachability problems of matrix semigroups. We prove that the satisfiability problems for □∨impi, ◊□∨impi and □◊∨impi with atomic propositions pi are decidable. This result solves the open problem of Li and Ying 2014. Notably, the decidability of □◊p can be interpreted as a generalization of Skolem-Mahler-Lech's celebrated theorem based on additive number theory. This paper's last part shows how the quantum finite automaton can model the general concurrent quantum programs, which may involve an arbitrary classical control flow.

USING IFS TO REVEAL BIASES IN THE DISTRIBUTION OF PRIME NUMBERS

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias.

On the Exponential Diophantine Equation 2^x-99^y=z^2

Abstract: In this work, we determine all non-negative integers solution to the exponential Diophantine equation . The mathematical process bases on several theorems in Number theory, including the modular arithmetic, Divisibility and the Division Algorithm. The solution set to the equation is .

Computing superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group

In algebraic geometry or number theory, enumerating or finding superspecial curves in positive characteristic p is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group. By executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for all primes p with 19≤p<500, and in finding a single one for all primes p with 19≤p<7000.

Smoothed asymptotics: From number theory to QFT

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultraviolet regularization scheme for loop integrals in quantum field theory which we call η regularization. This reveals a connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularization of loop integrals in quantum field theory. In particular, we note that a method for regularizing the series of natural numbers so that it converges to minus one-twelfth inspires a regularization scheme for non-Abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarization tensor and other higher point functions. We also comment on a possible connection to Schwinger proper time integrals. Published by the American Physical Society 2024

A new bound for the orthogonality defect of HKZ reduced lattices

Hermite–Korkin–Zolotarev (HKZ) reduction is an important notion of lattice reduction which plays a significant role in number theory (particularly the geometry of numbers), and more recently in coding theory and post-quantum cryptography. In this work, we determine a sharp upper bound on the orthogonality defect of HKZ reduced bases up to dimension 3. Using this result, we determine a general upper bound for the orthogonality defect of HKZ reduced bases of arbitrary rank. This upper bound is sharper than existing bounds in literature, such as the one determined by Lagarias, Lenstra and Schnorr [3].

Excursions in Number Theory, Algebra, and Analysis By Kenneth Ireland and Al Cuoco

Excursions in Number Theory, Algebra, and Analysis By Kenneth Ireland and Al Cuoco

In Memoriam: Peter Benjamin Borwein (1953–2020)

Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.

Arbitrary finite intersections of doubling measures and applications

We make major progress on a folkloric conjecture in analysis by constructing a measure on the real line which is doubling on all n-adic intervals for any finite list of n∈N, yet not doubling overall. In particular, we extend previous results in the area, including those of Boylan-Mills-Ward and Anderson-Hu, by using a wide array of substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, Ap weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.