All Questions
Tagged with integer-programming diophantine-equations
15 questions
1
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System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
- 25.4k
2
votes
1
answer
179
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Only trivial solution to a pair of constrained linear diophantine equations
Given positive integer $n$, we are looking for a set
of $n$ positive integers $a_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
$0 \le x_i \le \frac{...
- 25.4k
0
votes
2
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427
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Fastest way to solve non-negative linear diophantine equations
Let $A$ be a matrix in $M_{n \times m}(\mathbb{Z}_{\ge 0})$ without zero column. Let $V$ be a vector in $\mathbb{Z}_{> 0}^m$.
Question: What is the fastest way to find all the solutions $X \in \...
5
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240
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Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$
Crossposted at Theoretical Computer Science SE
A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
- 123
6
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410
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Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$
Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that
$$a_1 b_1 + \dots + a_n b_n = 1$$
has a solution in integers $b_1, \dots, b_n$.
I would like to get a bound saying ...
- 4,164
2
votes
1
answer
202
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Feasibility of constrained multivariable diophantine equations
Let $d$ be day, $m$ be month and $y$ be year fields of a date. I want to find few dates of format
$$(d^2\, mod\,\, 2 + (my + d^3) \,mod \,4) = 2$$
Is there a method to solve this type of equation or ...
- 131
6
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1
answer
405
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Existence of nonnegative solution in diophantine linear equations system with non negative coefficents
Given a Diophantine system of linear equations $Ax = b$, where $A \in \mathbb{N}^{m\times n}$ and $b\in\mathbb{N}^{m}$, is there a method to determine whether there exists a nonnegative solution $x\...
- 61
2
votes
0
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92
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On design of a (preferrably unimodular) matrix
Assume each entry is in $\Bbb Z$.
Say we want to solve $Ax=b$ where known $A$ is $n\times n$, unknown $x$ is $n\times1$ and $b$ is $n\times1$.
The absolute value of minors of augmented matrix $[A|b]$...
- 13.9k
1
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1
answer
116
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Sign Enumeration
What is the number of solutions of $(a_i)_{i=1}^n$ such that
$$\sum_{i=1}^nia_i\le b,\quad a_i\in\{-1,1\},\quad \sum_{i=1}^n{a_i}=c$$
given $b,c\in\mathbf Z$?
Is there a generating function solution?
- 2,239
3
votes
2
answers
608
views
Minimal solution of simultaneous congruences
I would to determine the set of values $\lbrace a_1,a_2,a_3,\ldots,a_n \rbrace$ that minimizes the value of $x$ such that:
$$x\equiv a_1\mod p_1$$
$$\vdots$$
$$x\equiv a_n\mod p_n$$
where every ...
- 31
2
votes
1
answer
366
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System of congruences
I have a system of $n$ congruences.
the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form:
$(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq m}}^n{\frac{x_iy_m}{p_ip_m}}+\...
- 21
3
votes
1
answer
387
views
Sufficient condition for solvability of linear diophantine system
I would like to know under what conditions does an integer solution exist to the under-determined linear system:
Ax = b. (without constraints)
Where A is m x n matrix with positive integers entries (...
- 31
1
vote
2
answers
752
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basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
- 51
6
votes
2
answers
2k
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is there a solution to this linear Diophantine system?
I have a matrix $A \in \mathbb Z^{n \times m}$, where $m > n$, and a vector $b \in \mathbb Z^n$. Under what conditions does
$$Ax = b$$
have an integer solution? Is there a way to bound the norm ...
- 61
3
votes
0
answers
3k
views
0,1 solution to system of linear integer equations
I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-matrix and $m$-vector of nonnegative integers (respectively).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
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