Diophantine System Research Articles

On elliptic curves assigned to a pair of right triangles with an equal hypotenuse

Consider a pair of right triangles with an equal hypotenuse. This turns out to solve the diophantine system of equations a 2 + b 2 = c 2 + d 2 = e 2 in integers. To this system we associate a family of elliptic curves with defining equation y 2 = (x−a 2)(x−b 2)(x−c 2)+a 2 b 2 c 2. We show that there exists a subfamily of rank ≥ 3 over ℚ(m, n, k, ℓ) and obtain a subfamily of rank (exactly) four over ℚ(k) and determine a set of its free generators. Besides, we show there exist infinitely many elliptic curves of rank ≥ 5 parameterized by a rank five quartic elliptic curve. We also find a few particular examples with higher ranks. The families we construct have ℤ/2ℤ torsion subgroups in general. The previous work [13] has obtained similar Pythagorean quadruplet elliptic curve families in two parameters with rank ≥ 3. (Recall that by a Pythagorean quadruplet (a, b, c, d), we mean an integer solution to the quadratic equation a 2 + b 2 = c 2 + d 2.)

An Effective Algorithm for Solving Weak Fuzzy Complex Diophantine Equations in Two Variables

Weak fuzzy complex numbers are defined as , with as an extension of real numbers with . This paper is dedicated to studying weak fuzzy complex linear Diophantine equations in two weak fuzzy complex variables, by transforming the weak fuzzy complex Diophantine equation to a classical equivalent Diophantine system and going directly from the solutions of classical system into the desired equation. Algorithms for generating solutions of the previous equation will be presented in terms of theorems with many related examples that clarify the validity of our work.

On the variance of the Fibonacci partition function

TextWe determine the order of magnitude of the variance of the Fibonacci partition function. The answer is different to the most naive guess. The proof involves a diophantine system and an inhomogeneous linear recurrence. VideoFor a video summary of this paper, please visit https://youtu.be/nTeO_6Chqbo.

Rational right triangle tripleswith special linear relationship of areas and perimeters

Suppose that the rational right triangle triple is $(T_1,T_2,T_3)$, their areas are $A_i~(i=1,2,3)$, and perimeters are $P_i~(i=1,2,3)$. By the theory of elliptic curves, we investigate the solvability of the following Diophantine system \[A_1+\alpha A_2=\beta A_3,\qquad P_1+\alpha P_2=\beta P_3,\]where $\alpha$ and $\beta$ are rational numbers. When $(\alpha,\beta)=(-2,-1)$ or $(\alpha,\beta)=(1,1)$, we show that there are infinitely many rational right triangle triples with the same perimeter and the areas in arithmetical progression or with the areas and perimeters satisfying the linear recurrence equation of Lucas sequence respectively. Moreover, we prove that there is no rational right triangle triple whose areas, perimeters and radii of the inscribed circles satisfy the linear recurrence equation of Lucas sequence respectively.

Integers of a quadratic field with prescribed sum and product

For given $k,\ell \in \mathbb Z$ we study the Diophantine system $$x+y+z=k, \quad x y z = \ell $$ for $x,y,z$ integers in a quadratic number field, which has a history in the literature. When $\ell =1$, we describe all such solutions; only for $k=5,6$, do

Efficient reliability computation of a multi-state flow network with cost constraint

The reliability and cost integrated performance measure R(d,b) of a multi-state flow network, defined as the probability of sending a flow of d units from the source to the sink with the total transmission cost no more than b, can be computed by means of (d, b)-minimal paths ((d, b)-MPs). The existing methods search for (d, b)-MPs by solving a large Diophantine system or several Diophantine subsystems that are shown to be NP-hard, then the computational efforts are prohibitive. This paper proposes a new search method for (d, b)-MPs, and major contributions include: (1) a correlation between (d, b)-MPs and minimum cost circulations is established, enabling the solution of (d, b)-MPs to be accomplished via minimum cost circulations; (2) a distinctive method merging the well-known capacity scaling algorithm and a decomposition technique is presented to find (d, b)-MPs. In contrast to the existing methods, the presented algorithm seeks for (d, b)-MPs without depending upon the solutions of Diophantine system and generating any duplicate (d, b)-MPs. An illustration of the proposed algorithm is presented, and computational results indicate the advantage of our algorithm over the existing methods.

Computing the reliability of a stochastic distribution network subject to budget constraint

Reliability and delivery cost are two crucial indicators to show whether a distribution network is in the stable and high-quality operation. This paper develops a (d, b)-minimal path ((d, b)-MP) based algorithm to compute the reliability R(d,b) of a stochastic distribution network as the probability that a prescribed demand d can be successfully delivered from the source to the destination subject to the constraint that the total delivery cost is not above b. Specifically, this paper focuses two aspects to facilitate the solution of (d, b)-MPs. Firstly, by transforming a large Diophantine system into several small Diophantine systems, a modified model is constructed to promote the efficiency of searching for (d, b)-MP candidates and narrow the scope of duplicate (d, b)-MPs. Secondly, a novel technique is devised to identify all duplicate (d, b)-MPs, and its efficiency advantage is theoretically proven. Finally, it is demonstrated through numerical experiments that the developed algorithm compares favorably with the existing ones in the literature, and a case study is provided to display the application of the algorithm.

English

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{{\matrix{{n = {a_1} + {a_2} + \ldots + {a_{s - 1}},} \hfill \cr {{a_1}{a_2} \ldots {a_{s - 1}}({a_1} + {a_2} + \ldots + {a_{s - 1}}) = {b^s}} \hfill \cr}} \right.$$ has positive integer or rational solutions n, b, aj, i = 1, 2, …, s − 1, s ⩾ 3. Using the theory of elliptic curves, we prove that it has no positive integer solution for s = 3, but there are infinitely many positive integers n such that it has a positive integer solution for s ⩾ 4. As a corollary, for s ⩾ 4 and any positive integer n, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that it has infinitely many positive rational solutions for s ⩾ 4 and a fixed positive integer n.

Finding all multi-state minimal paths of a multi-state flow network via feasible circulations

Multi-state minimal paths, called d-minimal paths (d-MPs), are a popular tool for computing the reliability of multi-state flow networks. Most of the existing algorithms seek for d-MPs by solving an NP-hard Diophantine system in terms of the implicit enumeration method that is simple yet inefficient. This paper focuses on the development of a novel method for finding all d-MPs. By exploring the relationship between d-MPs and feasible circulations that are a well-known network flow problem, this paper proposes a distinctive method integrating the traditional max-flow algorithm and a partition technique to find all d-MPs. The developed method seeks for d-MPs by solving feasible circulations, rather than by solving an NP-hard Diophantine system; additionally, it neither requires MPs nor generates duplicate d-MPs during the solution process. Both theoretical and experimental results demonstrate the advantage of the developed method, and a real delivery network example is presented to illustrate its application.

Arithmetic properties of polynomials

First, we prove that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q) \end{aligned}$$ has infinitely many integer solutions for $$f(X)=X(X+a)$$ with nonzero integers $$a\equiv 0,1,4\pmod {5}$$ . Second, we show that the above Diophantine system has an integer parametric solution for $$f(X)=X(X+a)$$ with nonzero integers a, if there are integers m, n, k such that $$\begin{aligned} {\left\{ \begin{array}{ll} (n^2-m^2)(4mnk(k+a+1)+a(m^2+2mn-n^2)) &{}\equiv 0\pmod (m^2+2mn-n^2)((m^2-2mn-n^2)k(k+a+1)-2amn) &{}\equiv 0\pmod {(m^2+n^2)^2}, \end{array}\right. } \end{aligned}$$ where $$k\equiv 0\pmod {4}$$ when a is even, and $$k\equiv 2\pmod {4}$$ when a is odd. Third, we get that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)=\frac{f(r)}{f(s)} \end{aligned}$$ has a five-parameter rational solution for $$f(X)=X(X+a)$$ with nonzero rational number a and infinitely many nontrivial rational parametric solutions for $$f(X)=X(X+a)(X+b)$$ with nonzero integers a, b and $$a\ne b$$ . Finally, we raise some related questions.

Verification of hypertorus communication grids by infinite petri nets and process algebra

A model of a hypertorus communication grid has been constructed in the form of an infinite Petri net. A grid cell represents either a packet switching device or a bioplast cell. A parametric expression is obtained to allow a finite specification of an infinite Petri net. To prove properties of an ideal communication protocol, we derive an infinite Diophantine system of equations from it, which is subsequently solved. Then we present the programs htgen and ht-mcrl2-gen, developed in the C language, which generate Petri net and process algebra models of a hypertorus with a given number of dimensions and grid size. These are the inputs for the respective modeling tools Tina and mCRL2, which provide model visualization, step simulation, state space generation and reduction, and structural analysis techniques. Benchmarks to compare the two approaches are obtained. An ad-hoc induction-like technique on invariants, obtained for a series of generated models, allows the calculation of a solution of the Diophantine system in a parametric form. It is proven that the basic solutions of the infinite system have been found and that the infinite Petri net is bounded and conservative. Some remarks regarding liveness and liveness enforcing techniques are also presented.

Generating Functions of Weighted Voting Games, MacMahon’s Partition Analysis, and Clifford Algebras

MacMahon’s Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighte...

On a problem of Pethő

In this paper we deal with a problem of Pethő related to existence of a quartic algebraic integer α for whichβ=4α4α4−1−αα−1 is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many α's such that the corresponding β is quadratic. Moreover, we present a description of quartic numbers α such that the corresponding β is a quadratic real number.

Finding all the Lower Boundary Points in a Multistate Two-Terminal Network

System reliability of a multistate flow network can be computed in terms of all the lower boundary points, called d-minimal paths (d-MPs). Although several algorithms have been proposed in the literature for the d-MP problem, there is still room for improvement upon its solution. Here, some new results are presented to improve the solution of the problem. A simple novel algorithm is improved to solve a Diophantine system that appeared in the d-MP problem. Then, an improved algorithm is proposed for the d-MP problem. It is also explained how the proposed algorithm can be used in order to assess the reliability of some smart grid communication networks. We provide the complexity results and show the main algorithm to be more efficient than the existing ones in terms of execution times through some benchmark networks and a general network example. Moreover, we compare the algorithms through one thousand randomly generated test problems using the Dolan and More's performance profile.

On the Diophantine system $$f(z)\,{=}\,f(x)f(y)\,{=}\,f(u) f(v)$$ f ( z ) = f ( x ) f ( y ) = f ( u ) f ( v )

We show that the Diophantine system $$\begin{aligned} f(z)=f(x)f(y)=f(u)f(v) \end{aligned}$$ has infinitely many nontrivial positive integer solutions for \(f(X)=X^2-1\), and infinitely many nontrivial rational solutions for \(f(X)=X^2+b\) with nonzero integer b.

From Feynman rules to conserved quantum numbers, II

Extending the basic formalism described in a closely related article, this paper defines the reduced equations of a graph model and presents various theoretical results involving those equations, both diophantine and modular. The reduced equations contribute to clarify the existence (or non-existence) of certain types of identities, for some classes of models (a number of which is analysed in the second part of this paper).It is also pointed out that there exist systems of equations (related to the reduced equations) that can replace the system of constituent equations for the purpose of deriving the identities of a given model. A definite diophantine system is proposed, one for which the coefficient matrix is typically smaller than that of the system of constituent equations.

Sparse Solutions of Linear Diophantine Equations

We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel's Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

Almost prime solutions to diophantine systems of high rank

Let [Formula: see text] be a family of [Formula: see text] integral forms of degree [Formula: see text] and [Formula: see text] be a family of pairwise linearly independent linear forms in [Formula: see text] variables [Formula: see text]. We study the number of solutions [Formula: see text] to the diophantine system [Formula: see text] under the restriction that [Formula: see text] has a bounded number of prime factors for each [Formula: see text]. We show that the system [Formula: see text] that has the expected number of such “almost prime” solutions under similar conditions was established for existence of integer solutions by Birch.

A New Algorithm for Generating All Minimal Vectors for the <formula formulatype="inline"> <tex Notation="TeX">$q$</tex> </formula> SMPs Reliability Problem With Time and Budget Constraints

The quickest path reliability problem evaluates the probability of transmitting some given units of flow from a source node to a sink node through a single minimal path in a stochastic-flow network within some specified units of time. This problem has been recently extended to the case of transmitting flow through $q$ separate minimal paths (SMPs) simultaneously within a budget constraint. Here, we propose a new algorithm to solve such problems. The algorithm finds all the minimal vectors for which a given demand level $d$ units of flow can be transmitted through $q$ SMPs from a source node to a sink node satisfying some given time and budget limits. In our proposed algorithm, a special Diophantine system is needed to be solved, for which we provide a particularly efficient procedure with linear time complexity in terms of the number of its solutions. The complexity results are derived to demonstrate the efficiency of our proposed algorithm in comparison with others. To illustrate the practical efficiency of the algorithm, a network example is considered to compare our algorithm with a recently proposed one. Moreover, a comparative computational experiment is conducted on an extensive collection of random test problems using the performance profile of Dolan and More.

A Euclid style algorithm for MacMahon's partition analysis

Solutions to a linear Diophantine system, or lattice points in a rational convex polytope, are important concepts in algebraic combinatorics and computational geometry. The enumeration problem is fundamental and has been well studied, because it has many applications in various fields of mathematics. In algebraic combinatorics, MacMahon's partition analysis has become a general approach for linear Diophantine system related problems. Many algorithms have been developed, but “bottlenecks” always arise when dealing with complex problems. While in computational geometry, Barvinok's important result asserts the existence of a polynomial time algorithm when the dimension is fixed. However, the implementation by the LattE package of De Loera et al. does not perform well in many situations. By combining excellent ideas in the two fields, we generalize Barvinok's result by giving a polynomial time algorithm for MacMahon's partition analysis in a suitable condition. We also present an elementary Euclid style algorithm, which might not be polynomial but is easy to implement and performs well. As applications, we contribute the generating series for magic squares of order 6.