Theorem In Number Theory Research Articles

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.

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 .

Research and application of Chinese remainder theorem

The Chinese remainder theorem (denoted it as " the theorem" in this article) was originally an important theorem in number theory. It played a vital role in the integer solution of the congruence equation in ancient times. With the continuous development of the algebraic system, the theorem naturally has different forms. This paper will show some research and applications based on the theorem. For example, the theorem in polynomial form, the theorem in the form of group theory, the theorem on unitary rings, the theorem on polynomial ring modules, etc. It is not difficult to know that integers and polynomials are special rings, so this the two forms of the theorem are the theorems that can be covered on the unitary ring. In fact, the theorem in the form of group theory is also covered. This paper will elaborate the first three forms of the theorem and give their specific applications.

On the Exponential Diophantine Equation 2x+15y=z2

In this article, we solve the exponential Diophantine equation 2 2 15 x y + = z where x y, and z are non-negative integers. The basic theorems in Number theory are given and applied to find all solutions. The result reveals that there are only three solutions to the equation.

A Survey on Newmann’s proof of the prime number theorem

The prime number theorem is one of the most important theorems in number theory, which tells the asymptotic behavior of positive prime numbers. It was first proved independently by Jacques Hadamard and Charles-Jean de la Valĺee Poussin in 1896, by using the Riemann zeta function as a tool. We summarize Newmann’s proof in this survey.

Division algorithm and Kronecker’s approximation theorem

Using division algorithm and basic notions of convergence of sequences in real–line, we prove that a real number $$\theta$$ is irrational if and only if there is an eventually nonconstant sequence $$\{p_n\theta +q_n\}$$ converging to 0, where $$p_n$$ and $$q_n$$ are integers for each natural number n. This approach leads to alternative proofs of weaker versions of the classical Dirichlet and Kronecker approximation theorems in number theory.

On the asymptotic idealness of the Asmuth-Bloom threshold secret sharing scheme

The Chinese remainder theorem (CRT) is a fundamental theorem in number theory, widely used in cryptography to design secret sharing schemes. The CRT-based secret sharing schemes proposed so far make use of sequences of pairwise co-prime integers with special properties. The way these sequences are chosen plays a crucial role in the security achieved by the schemes that rely on them. Moreover, the CRT-based secret sharing schemes could achieve at most asymptotic idealness. In this paper we prove that the Asmuth-Bloom threshold secret sharing scheme is asymptotic ideal if and only if it is based on 1-compact sequences of co-primes. Apart from this, a comprehensive analysis of the known variants of the Asmuth-Bloom threshold secret sharing scheme is provided, clarifying the security properties achieved by each of them.

Some variants of Lagrange's four squares theorem

Lagrange's four squares theorem is a classical theorem in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and obtain various refinements of Lagrange's theorem. We show that any nonnegative integer can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ (or $x+y+z+2w$, or $x+2y+3z+w$) a square (or a cube). Also, every $n=0,1,2,\ldots$ can be represented by $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+3z$ (or $x+2y+3z$) a square (or a cube), and each $n=0,1,2,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $(10w+5x)^2+(12y+36z)^2$ (or $x^2y^2+9y^2z^2+9z^2x^2$) a square. We also provide an advance on the 1-3-5 conjecture of Sun. Our main results are proved by a new approach involving Euler's four-square identity

Congruence subgroups from representations of the three-strand braid group

Ng and Schauenburg proved that the kernel of a (2+1)-dimensional topological quantum field theory representation of SL(2,Z) is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group B3 with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator σ1 is between 2 and 5 the kernel projects onto a congruence subgroup of PSL(2,Z) and compute its level. However, we prove that for three dimensional representations, the projective order is not enough to decide the congruence property. For each integer of the form 2ℓ≥6 with ℓ odd, we construct a pair of non-congruence subgroups associated with three-dimensional representations having finite image and σ1 mapping to a matrix with projective order 2ℓ. Our technique uses classification results of low dimensional braid group representations, and the Fricke–Wohlfahrt theorem in number theory.

The Green’s function for the Hückel (tight binding) model

Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green’s function, G, of the N×N Hückel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d− dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green’s function and discuss applications related to transport and conductivity.

A Model-Based Approach for The Demonstration of Fermat’s Last Theorem

Model-based approaches have many applications in bioinformatics studies. Fermat’s last theorem is a theorem in number theory. This paper presented a model-based approach for the demonstration of Fermat’s Last Theorem (FLT). Through the establishment of mathematical model, Fermat’s Last Theorem (FLT) is converted into a problem for feasible solutions in linear programming. By prompting assumptions, and refuting the assumptions, it is demonstrated that Fermat’s Last Theorem is right. The model-based approach for the demonstration of Fermat's Last Theorem (FLT) can assist scientific researchers in biological researches.

A combinatorial proof of a decomposition property of reduced residue systems

In this paper, we look at three common theorems in number theory: the Chinese remainder theorem, the multiplicative property of the Euler totient function, and a decomposition property of reduced residue systems. We use a grid of squares to give simple transparent visual proofs.

Saints and Scoundrels and Two Theorems That Are Really the Same

SummaryThe Chinese remainder theorem and the polynomial interpolation theorem are foundational theorems of number theory and numerical analysis, respectively. These two theorems are special cases of a construct in a more general setting and we describe a scenario—namely, two back-to-back classes—in which students can discover this fact. We end by describing that general setting, in which the key idea is an ability to write 1 in a special way.

Asymptotic evaluations for some double sums in number theory

In this paper, we indicate a method to obtain asymptotic evaluations for double sums from the asymptotic evaluations of a single sum. As applications, we show that all results in the recent paper [New extensions of some classical theorems in number theory, J. Number Theory133(13) (2013) 3771–3795] can be used to obtain some asymptotic evaluations for double sums. Further, we give asymptotic evaluations for double sums which involve the divisor and the Euler totient function.

Cryptanalyzing a class of image encryption schemes based on Chinese remainder theorem

As a fundamental theorem in number theory, the Chinese Reminder Theorem (CRT) is widely used to construct cryptographic primitives. This paper investigates the security of a class of image encryption schemes based on CRT, referred to as CECRT. Making use of some properties of CRT, the equivalent secret key of CECRT can be recovered efficiently. The required number of pairs of chosen plaintext and the corresponding ciphertext is only (1+⌈(log2L)/l⌉), the attack complexity is only O(L), where L is the plaintext length and l is the number of bits representing a plaintext symbol. In addition, other defects of CECRT, such as invalid compression function and low sensitivity to plaintext, are reported. The work in this paper will help clarify positive role of CRT in cryptology.

Stable birational equivalence and geometric Chevalley-Warning

We propose a “geometric Chevalley-Warning” conjecture, that is, a motivic extension of the Chevalley-Warning theorem in number theory. Its statement is equivalent to a recent question raised by F. Brown and O. Schnetz. In this paper we show that the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in P 3 \mathbb {P}^3 . The last section is devoted to verifying the conjecture for certain special kinds of hypersurfaces of any dimension. As a by-product, we obtain information on the Grothendieck classes of the affine “Potts model” hypersurfaces considered by Aluffi and Marcolli.

New extensions of some classical theorems in number theory

In this paper we show that for k ∈ N ∪ { 0 } , under natural assumptions on the functions g and h , for a large class of Riemann integrable functions f : [ 0 , 1 ] k + 1 → R (not all, for k ∈ N ; and all, for k = 0 ), the following equality holds lim x → ∞ 1 h ( x ) ∑ n ⩽ x f ( n x , ln 1 n ln 1 x , … , ln k n ln k x ) g ( n ) = ∫ 0 1 f ( x , 1 , … , 1 ︸ k - times ) d x . Using these results for prime numbers, we obtain some new extensions of the classical version from 1917 Polyaʼs theorem in number theory.

On Griffiths–Lang's conjecture and related topics

We mainly give a brief survey on some methods and results related to Griffiths–Lang's conjecture on the second main theorem of holomorphic curves type into projective varieties. Also presented is the analogue of Cartan's theorem in number theory by Cartan's method, which is a refined form of Schmidt's subspace theorem.

Gene expression from 2-adic dynamical systems

We perform geometrization of genetics by representing genetic information by points of the 4-adic information space. By a well-known theorem of number theory this space can also be represented as the 2-adic space. The process of DNA reproduction is described by the action of a 4-adic (or equivalently 2-adic) dynamical system. As we know, the genes contain information for production of proteins. The genetic code is a degenerate map of codons to proteins. We model this map as the functioning of a polynomial dynamical system. The purely mathematical problem under consideration is to find a dynamical system reproducing the degenerate structure of the genetic code. We present one of possible solutions of this problem.

Quantum properties of irrational triangular billiards

Triangles with sides given by consecutive integers (N , N+1 , N+2) are fully irrational (all angles irrational with pi) if 3<N<infinity . Rational approximations to their angles and the Hurwitz theorem in number theory are used to define a parameter h that quantifies the irrationality of each triangle. The energy level statistics [spacing distribution p(s) and spectral rigidity Delta(3)(L)] of quantum billiards from this one-parameter family of triangles are investigated. The behavior of h with varying N and the numerically calculated level dynamics are found to be closely related: h exhibits a local maximum at N=10, around which agreement with Gaussian orthogonal ensemble (GOE) spectral fluctuations is observed. As N is increased, h decreases and the statistics depart from GOE. Structures appear in p(s) for N>120 and eventually the occurrence of gaps in the distribution for N ~ 180 define the onset of a long crossover towards the sequence observed in the integrable limit of the equilateral triangle (N-->infinity) .