Integer Algorithms to Solver Diophantine Linear Equations and Systems

The aim of this paper is to prove the possibility of linearization of such equations by means of introduction of new variables. For $n=2$ such a procedure is well known, when new variables are components of spinors and they are widely used in mathematical physics. For example, parametrization of Pythagoras threes $a^{2} +b^{2}$, $a^{2} -b^{2}$, $2ab$ may be cited as an example in number theory where two independent variables form a spinor which can be obtained by solution of ...

Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local to global type statement is shown via classical Hardy-Littlewood type methods which provides sufficient conditions for the solubility of $\mathfrak{p}(\mathbf{x})=\mathbf{s}$ under the condition that each of the $x_i$'s is prime.

The goal of the work is to take on and study one of the fundamental tasks studying Diophantine n-gons (the author of the paper considers an integral n-gon is Diophantine as far as determination of combinatorial properties of each of them requires solution of a certain Diophantine equation (equation sets)).

In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.

It is a generalization of Pell's equation $x^2-Dy^2=0$. Here, we show that: if our Diophantine equation has a particular integer solution and $ab$ is not a perfect square, then the equation has an infinite number of solutions; in this case we find a close expression for $(x_n,y_n)$, the general positive integer solution, by an original method. More, we generalize it for any Diophantine equation of second degree and with two unknowns $f(x,y)=0$.

We deal with the problem to find the number $P(b)$ of integer non-negative solutions of an equation $\sum_{i=1}^{n} a_i x_i=b$, where $a_1,a_2,...,a_n$ are natural numbers and $b$ is a non-negative integer. As different from the traditional methods of investigation of the function $P(b)$, in our study we do not employ the techniques of number series theory, but use in the main the properties of the Kronecker function and the elements of combinatorics. The formula is derived t...

In the paper I considered methods for solving equations of the form axb+cxd=e in the algebra which is finite dimensional over the field.

An algorithm which either finds an nonzero integer vector ${\mathbf m}$ for given $t$ real $n$-dimensional vectors ${\mathbf x}_1,...,{\mathbf x}_t$ such that ${\mathbf x}_i^T{\mathbf m}=0$ or proves that no such integer vector with norm less than a given bound exists is presented in this paper. The cost of the algorithm is at most ${\mathcal O}(n^4 + n^3 \log \lambda(X))$ exact arithmetic operations in dimension $n$ and the least Euclidean norm $\lambda(X)$ of such integer v...

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace~$L$ in $R^n$ is a free $R$-module of rank~$f=n-r$. We construct a free submodule $M$ of $L$ of rank~$f$ naturally associated to $A$ and whose basis is easily computable, we determine t...

For a positive integer n, let {\theta}(n) denote the smallest positive integer b such that for each system S \subseteq {x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}} which has a solution in positive integers x_1,...,x_n and which has only finitely many solutions in positive integers x_1,...,x_n, there exists a solution of S in ([1,b] \cap N)^n. We conjecture that there exists an integer {\delta} \geq 9 such that the inequality {\theta}(n) \leq (2^{2^{n-5}}-1)^{2^{n-5}}+1...