Integer Solutions of Linear Diophantine Equations Form a Group
نویسندگان
چکیده
منابع مشابه
Small solutions of linear Diophantine equations
Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi <...
متن کاملSparse Solutions of Linear Diophantine Equations
We present structural results on solutions to the Diophantine system Ay = b, y ∈ Z ≥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.
متن کاملOptical solutions for linear Diophantine equations
Determining whether a Diophantine equation has a solution or not is the most important challenge in solving this type of problems. In this paper a special computational device which uses light rays is proposed to answer this question, namely check the existence of nonnegative solutions for linear Diophantine equations. The way of representation for this device is similar to an directed graph, h...
متن کاملDiophantine Equations Related with Linear Binary Recurrences
In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...
متن کاملBounds on Positive Integral Solutions of Linear Diophantine Equations
Assuming the existence of a solution, we find bounds for small solutions x of the finite matrix equation Ax = B, where each entry of A, B is an integer, and x is a nontrivial column vector with nonnegative integer entries. 0. Introduction. In [1], [5] and [6] there arise in a topological setting, systems of linear equations with integer coefficients. The problem is to find a bound K depending o...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Missouri Journal of Mathematical Sciences
سال: 2006
ISSN: 0899-6180
DOI: 10.35834/2006/1802135