If I have a solution to say $$4x_1+3x_2+3x_3+3x_4...+3x_n=4y$$ where $x_i ,y$ are non negative integers can I generate other solutions from my initial solution? And if I can would all solutions follow from this initial one?

  • $\begingroup$ what I am really trying to get at is if my initial solution contains all even integers then would solutions generated contain at least one even integer because the coefficient in front of $x_1$ is even $\endgroup$ – argamon May 15 '20 at 0:40
  • $\begingroup$ Is it a linear equation you are interested in? If yes, you sould find more solution by finding integer vectors in the "kernel" of this system. Here, it is the space orthogonal to $(4, 3, 3, \dotsc, 3)$ or more precisely the integer lattice living in this orthogonal. $\endgroup$ – KeiOh May 15 '20 at 2:15
  • $\begingroup$ For your comment, I am not sure it works: take $n=5$, and consider $u=(-3,1,1,1,1)$. Then $4u_1 + 3u_2 +\dotsb + 3u_5 = 0$. Now look at the equation $4x_1 + 3x_2 + \dotsb + 3x_5 = 8$. Then $x = (2, 0, \dotsc, 0)$ is a solution, but then so is $x+u$, isn't it? But $x+u$ as all odd coordinates. $\endgroup$ – KeiOh May 15 '20 at 2:19

I'm not sure what exactly are you asking for, but I observed the following properties.

I'm assuming you want solutions for $(x_1,\dots,x_n)$ given $y$. Let $z=y-x_1$, then $$3(x_2+x_3+x_4\dots+x_n)=4z$$

Hence, $z$ must be a multiple of $3$, so we write $3k=y-x_1\implies x_1=y-3k$ for $0\le k\le\left\lfloor\frac{y}{3}\right\rfloor$.

If $x_1=y-3k$ for some $0\le k\le\left\lfloor\frac{y}{3}\right\rfloor$, then $x_2,\dots,x_n$ are a solution $\iff\sum_{i=2}^n x_i = 4k$.

Now to get all solutions, simply iterate all $k$, and for each, all $x_2,\dots,x_n$ s.t. $\sum_{i=2}^n x_i = 4k$.

That is, you can generate a solution form some other in two ways ("one two-case operation"):

  1. (We redistribute the sum condition:) You can decrease (increase) some component $x_i,i\ge 2$ by $1$ as long as you increase (decrease) some other component $x_{j},j\ge 2$.

  2. (We change the case of $k$:) You can decrease (increase) $x_1$ by $3$, as long as you increase (decrease) up to four other components $x_i,x_j,x_t,x_l$ for $2\le i \le j \le t \le l$ by $1$.

The set of all solutions is closed relative to this "two-case operation", so by repeating these two cases on any solution and all subsequently generated solutions, you can eventually reach all solutions.


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