A Class of Linear Diophantine Equations which Always has a Solution?
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Consider the set of linear Diophantine equations given by $Ax=p$ where $A$ is an $n times m$ matrix with non-negative integer elements, $x$ is a vector with non-negative integer elements and $p = (tilde{p} , tilde{p} , dots , tilde{p})$ for a positive integer $tilde{p}$. The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$.
Furthermore I have conditions on the matrix $A$. Denote the $i$'th row of $A$ by $a_i$. Then $a_i$ must be non-zero and cannot be of the form $a_i = a_j + (b_1 , b_2, dots , b_m)$ for $b_k geq 0$ for all $k$. This condition must hold for all $i$ and $j$ such that $ i neq j$.
Note that I do not require $n leq m$.
Question: Is it always possible to find $x$ and $tilde{p}$ for a given matrix $A$ satisfying my conditions? I believe the answer is yes, but I am finding this difficult to prove.
Any and all help is greatly appreciated!
number-theory linear-diophantine-equations
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
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add a comment |
$begingroup$
Consider the set of linear Diophantine equations given by $Ax=p$ where $A$ is an $n times m$ matrix with non-negative integer elements, $x$ is a vector with non-negative integer elements and $p = (tilde{p} , tilde{p} , dots , tilde{p})$ for a positive integer $tilde{p}$. The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$.
Furthermore I have conditions on the matrix $A$. Denote the $i$'th row of $A$ by $a_i$. Then $a_i$ must be non-zero and cannot be of the form $a_i = a_j + (b_1 , b_2, dots , b_m)$ for $b_k geq 0$ for all $k$. This condition must hold for all $i$ and $j$ such that $ i neq j$.
Note that I do not require $n leq m$.
Question: Is it always possible to find $x$ and $tilde{p}$ for a given matrix $A$ satisfying my conditions? I believe the answer is yes, but I am finding this difficult to prove.
Any and all help is greatly appreciated!
number-theory linear-diophantine-equations
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
$endgroup$
$begingroup$
"The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$." This is easy. Just take $x=0$ and $tilde{p}=0$. This works for all $A$.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 11:55
$begingroup$
My apologies, I should have been more specific. I am looking for nontrivial solutions. Alternatively I could say strictly positive $tilde{p}$.
$endgroup$
– Kooper de Lacy
Dec 22 '18 at 13:51
add a comment |
$begingroup$
Consider the set of linear Diophantine equations given by $Ax=p$ where $A$ is an $n times m$ matrix with non-negative integer elements, $x$ is a vector with non-negative integer elements and $p = (tilde{p} , tilde{p} , dots , tilde{p})$ for a positive integer $tilde{p}$. The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$.
Furthermore I have conditions on the matrix $A$. Denote the $i$'th row of $A$ by $a_i$. Then $a_i$ must be non-zero and cannot be of the form $a_i = a_j + (b_1 , b_2, dots , b_m)$ for $b_k geq 0$ for all $k$. This condition must hold for all $i$ and $j$ such that $ i neq j$.
Note that I do not require $n leq m$.
Question: Is it always possible to find $x$ and $tilde{p}$ for a given matrix $A$ satisfying my conditions? I believe the answer is yes, but I am finding this difficult to prove.
Any and all help is greatly appreciated!
number-theory linear-diophantine-equations
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
$endgroup$
Consider the set of linear Diophantine equations given by $Ax=p$ where $A$ is an $n times m$ matrix with non-negative integer elements, $x$ is a vector with non-negative integer elements and $p = (tilde{p} , tilde{p} , dots , tilde{p})$ for a positive integer $tilde{p}$. The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$.
Furthermore I have conditions on the matrix $A$. Denote the $i$'th row of $A$ by $a_i$. Then $a_i$ must be non-zero and cannot be of the form $a_i = a_j + (b_1 , b_2, dots , b_m)$ for $b_k geq 0$ for all $k$. This condition must hold for all $i$ and $j$ such that $ i neq j$.
Note that I do not require $n leq m$.
Question: Is it always possible to find $x$ and $tilde{p}$ for a given matrix $A$ satisfying my conditions? I believe the answer is yes, but I am finding this difficult to prove.
Any and all help is greatly appreciated!
number-theory linear-diophantine-equations
number-theory linear-diophantine-equations
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
edited Dec 22 '18 at 13:49
Kooper de Lacy
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
asked Dec 22 '18 at 7:22
Kooper de LacyKooper de Lacy
12
12
$begingroup$
"The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$." This is easy. Just take $x=0$ and $tilde{p}=0$. This works for all $A$.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 11:55
$begingroup$
My apologies, I should have been more specific. I am looking for nontrivial solutions. Alternatively I could say strictly positive $tilde{p}$.
$endgroup$
– Kooper de Lacy
Dec 22 '18 at 13:51
add a comment |
$begingroup$
"The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$." This is easy. Just take $x=0$ and $tilde{p}=0$. This works for all $A$.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 11:55
$begingroup$
My apologies, I should have been more specific. I am looking for nontrivial solutions. Alternatively I could say strictly positive $tilde{p}$.
$endgroup$
– Kooper de Lacy
Dec 22 '18 at 13:51
$begingroup$
"The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$." This is easy. Just take $x=0$ and $tilde{p}=0$. This works for all $A$.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 11:55
$begingroup$
"The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$." This is easy. Just take $x=0$ and $tilde{p}=0$. This works for all $A$.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 11:55
$begingroup$
My apologies, I should have been more specific. I am looking for nontrivial solutions. Alternatively I could say strictly positive $tilde{p}$.
$endgroup$
– Kooper de Lacy
Dec 22 '18 at 13:51
$begingroup$
My apologies, I should have been more specific. I am looking for nontrivial solutions. Alternatively I could say strictly positive $tilde{p}$.
$endgroup$
– Kooper de Lacy
Dec 22 '18 at 13:51
add a comment |
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$begingroup$
"The aim is to determine $x$ and $tilde{p}$ for a given matrix $A$." This is easy. Just take $x=0$ and $tilde{p}=0$. This works for all $A$.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 11:55
$begingroup$
My apologies, I should have been more specific. I am looking for nontrivial solutions. Alternatively I could say strictly positive $tilde{p}$.
$endgroup$
– Kooper de Lacy
Dec 22 '18 at 13:51