Number Of Integer Solutions Research Articles

Power savings for counting solutions to polynomial-factorial equations

Let P be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions n≤N to n!=P(x) which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is o(N). The proof uses techniques of Diophantine and Padé approximation.

On the restricted partition function via determinants with Bernoulli polynomials. II

Let r ≥ 1 be an integer, a = (a1, . . . , ar) a vector of positive integers, and let D ≥ 1 be a common multiple of a1, . . . , ar. We prove that if D = 1 or D is a prime number then the restricted partition function pa(n) := the number of integer solutions (x1, . . . , xr) to Pr j=1 ajxj = n, with x1 ≥ 0, . . . , xr ≥ 0, can be computed by solving a system of linear equations with coefficients that are values of Bernoulli polynomials and Bernoulli–Barnes numbers.

The Quadratic Diophantine Equations x^2− P(t)y^2− 2P′(t)x + 4P(t)y + (P′(t))^2− 4P(t) − 1 = 0

Let P := P(t) be a non square polynomial. In this paper, we consider the number of integer solutions of Diophantine equation
 
 E : x2− P(t)y2− 2P′(t)x + 4P(t)y + (P′(t))2− 4P(t) − 1 = 0.
 
 We derive some recurrence relations on the integer solutions (xn,yn) of E. In the last section, we consider the same problem over finite fields Fpfor primes p ≥ 5. Our main results are generaliations of previous results given by Ozcok and Tekcan (Ozkoc and Tekcan, 2010).

On the number of integer points in a multidimensional domain

Abstract We provide a new upper estimate for the modulus of the difference |Λ ∩ 𝓢| − voln(𝓢)/det Λ, where 𝓢 ⊂ ℝn is a set of volume voln(𝓢) and Λ ⊂ ℝn is a complete lattice with determinant det Λ. This result has an important practical application, for example, in estimating the number of integer solutions of an arbitrary system of linear and nonlinear inequalities.

A new method of solving certain quartic and higher degree diophantine equations

In this paper, we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be applied to some diophantine systems in five or more variables. Under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, two examples being a sextic equation in four variables and two simultaneous equations of degrees four and six in six variables. We also simultaneously obtain arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. We obtain these solutions without finding a curve of genus 0 or 1 on the variety defined by the equations concerned. It appears that there exist projective varieties on which there are an arbitrarily large number of rational points and which do not contain a curve of genus 0 or 1 with infinitely many rational points.

On the restricted partition function

For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ and its polynomial part.

New techniques for linear arithmetic: cubes and equalities

We present several new techniques for linear arithmetic constraint solving. They are all based on the linear cube transformation, a method presented here, which allows us to efficiently determine whether a system of linear arithmetic constraints contains a hypercube of a given edge length. Our first findings based on this transformation are two sound tests that find integer solutions for linear arithmetic constraints. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width. Experimental results confirm that our tests are superior on these instances compared to several state-of-the-art SMT solvers. We also discovered that the linear cube transformation can be used to investigate the equalities implied by a system of linear arithmetic constraints. For this purpose, we developed a method that computes a basis for all implied equalities, i.e., a finite representation of all equalities implied by the linear arithmetic constraints. The equality basis has several applications. For instance, it allows us to verify whether a system of linear arithmetic constraints implies a given equality. This is valuable in the context of Nelson–Oppen style combinations of theories.

On the quaternary form x2 + xy + 7y2 + z2 + zt + 7t2

For any positive integer [Formula: see text], we state and prove a formula for the number of solutions in integers of [Formula: see text]

Algorithms for Integer Factorization Based on Counting Solutions of Various Modular Equations

This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.

Density of integer solutions to diagonal quadratic forms

Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q = 0, which lie in a box with sides of length 2B, as B → ∞. The estimates obtained are completely uniform in the coefficients of the form, and become sharper as they grow larger in modulus.

The average number of solutions of the Diophantine equation U2+V2=W3and related arithmetic functions

For the number of integer solutions of the title equation, withW≤;x (x a large parameter), an asymptotics of the form Ax log x + Bx + O(x1/2 (log x)3 (loglog x)2) is established. This is achieved in a general setting which furnishes applications to some other natural arithmetic functions.

An algorithm for the biobjective integer minimum cost flow problem

In this paper, we study the single commodity flow problems, optimizing two objectives simultaneously, where the flow values must be integer values. We propose a method that finds all the efficient integer points in the objective space. Our algorithm performs two phases. In the first phase, all integer points on the efficient boundary are found and in the second phase, the efficient integer points that do not lie on the efficient boundary are calculated. In addition, we carry out a computational experiment showing that the number of efficient integer solutions that do not lie on the efficient boundary is greater than the number of integer solutions on the efficient boundary. Scope and purpose In many combinatorial optimization problems, the selection of the optimum solution takes into account more than one criterion. For example, in transportation problems or in network flows problems, the criteria that can be considered are the minimization of the cost for selected routes, the minimization of arrival times at the destinations, the minimization of the deterioration of goods, the minimization of the load capacity that would not be used in the selected vehicles, the maximization of safety, reliability, etc. Often, these criteria are in conflict and for this reason, a multiobjective network flow formulation of the problem is necessary. The solution to this problem is searched for among the set of efficient points. Although multiobjective network flow problems can be solved using the techniques available for the multiobjective linear programming problem, network-based methods are computationally better. The multicriteria minimum cost flow problem has already merited the attention of several authors and the case which has been considered in literature is that which has two objectives, where the continuous flow values are permissible. However, the integer case of the biobjective minimum cost flow problem has scarcely been studied. Whereas, in many real network flow problems, integer values on flow values are required. In this paper, we propose an approach to solve the biobjective integer minimum cost flow problem. An algorithm to obtain all efficient integer solutions of this problem is introduced. This method is characterized by the use of the classic resolution tools of network flow problems, such as the network simplex method. It does not utilize the biobjective integer linear programming methodology. Furthermore, the method does not calculate dominated solutions, so it is not necessary to incorporate tools to eliminate dominated solutions.

Reciprocity and Polynomial Properties for Even Flows and Potentials on Directed Graphs

We consider special types of mod-λ flows, called odd and even mod-λ flows, for directed graphs, and prove that the numbers of such flows can be interpolated by polynomials in λ with the degree given by the cycle rank of the graph. The proofs involve computation of the number of integer solutions in a polyhedral region of Euclidean space using theorems due to Ehrhart. The resulting reciprocity properties of the interpolating polynomials for even flows are considered. The analogous properties of odd and even mod-λ potential differences and their associated potentials are also developed.

System of inequalities

Let Nk,n,r(P) be a number of integer solutions of the system of inequalities The main result is the following estimate for k − n2/4≫ nr log r This estimate has the right order with respect to P. For r=n this is the classical Vinogradov mean value theorem.

INDUCED EXTREMAL SURFACES

Under general assumptions on the functions and it is proved that the inequality where is the distance from to the nearest integer and , , has only a finite number of solutions in integers for almost all . This establishes the extremality of the surface .Bibliography: 11 titles.

On the number of solutions to a diophantine equation

Let A 1, …, A r, x 1, …, x r , and A be known positive integers. Let f( A) be the number of integer solutions ( x 1, …, x r ) satisfying the Diophantine equation ∑ j=1 r A jx j = A and the conditions 0 ⩽ x i ⩽ x j , j = 1, …, r. This paper expresses f( A) recursively as a linear function of f(0), f(1), …, f( A−1).

Simultaneous asymptotic diophantine approximations to a basic of a real cubic number field

The units in cubic number fields together with the uniform distribution theorem are used to prove the following theorem. Let 1, β 1, β 2 be a basis for a real cubic number field. Let C > 0 be a given constant. Let λ B equal the number of solutions in integers q, p 1, p 2 of the inequalities 0 < qβ i − p i < C q 1 2 ( i = 1, 2), 1 ≤ q ≤ B. Then λ B = O(1) or there is a constant C′ > 0 such that λ B ∼ C′ log B ( B → ∞). The stumbling blocks for generalizing this result to higher dimensions are discussed.

Simultaneous asymptotic Diophantine approximations

Let θ 1 , …, θ k be k real numbers. Suppose ψ( t ) is a positive decreasing function of the positive variable t . Define λ( N ), for all positive integers N , to be the number of solutions in integers p 1 …, p k , q of the inequalities and

ASYMPTOTIC DIOPHANTINE APPROXIMATIONS TO E

—Schmidt5 proved that for almost all real numbers α, the number of solutions in integers p, q of the inequalities $$ \left| {q\alpha - p} \right| < 1/qand1\underline \leqslant q\underline \leqslant B$$ is asymptotic to a constant times log B. One might conjecture that the classical numbers (e.g., algebraic numbers, e, π) behave like almost all numbers. Machine computations’ were carried out for some of these numbers, and they seemed to bear out such a conjecture. Also, Lang3 has proved that the estimate is valid when α is a real quadratic irrationality.

Note on the Hardy-Landau Summation Formula

Broadly speaking, the Hardy-Landau summation formula * is given bywhere r(n) is the number of integer solutions of the Diophantine equation x2 + y2 = n, and f(x) and g(x) are transforms with respect to the Watson kernel that is:and