Does this involve solving a linear diophantine equation?

Asked Jun 18 '17 at 06:30

Active Jun 18 '17 at 06:30

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I'm not sure whether this is strictly a question about solving diophantine equations...

Consider the linear diophantine equation: $$a_1x_1+a_2x_2+\cdots+a_kx_k = d$$ I know a solution to the equation exists and I know the values of $a_1, ... a_k$. I also know that $\gcd(a_1,a_2,\ldots,a_k)=2$.

Suppose I know the value of one (and only one) of the variables, ie $x_1$. With this information, is it possible to determine possible values for $d$?

One specific example is: $$54x_1 - 40x_2 + 28x_3 - 18x_4 + 10x_5 - 4x_6 = d$$ In this case, if $x_1=3$, can I determine possible values for $d$? I'm not particularly interested in solving for the other variables.

asked Jun 18 '17 at 06:30

Molonglo

1

possible $d$ has the form $a_1x_1 + \gcd(a_2,\ldots,a_k) s$ for any $s \in \mathbb{Z}$. – achille hui Jun 18 '17 at 07:36
Thank you. So, if $gcd(a_2,...,a_k)$ also equals $2$, there would be solutions for any even $d$. – Molonglo Jun 18 '17 at 08:30
Yes. if $\gcd(a_2,\ldots,a_k)$ is $2$, every even $d$ is possible (assume $x_2,\ldots,x_k$ can be negative). – achille hui Jun 18 '17 at 08:33

Does this involve solving a linear diophantine equation?

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