A Linear Diophantine Equation and its solutions

A Linear Diophantine Equation, as you might guess, is a equation that starts with three fixed numbers, a,b and c, and a pair of numbers that can change, x and y such that:

And as you might see at a glance, both x and y graphed on a plane as coordinates will make up a straight line where the equation holds. But really what are the solutions for it? To find that out we must first remember the fact that the gcd(a,b) can be calculated similar to a Linear Diophantine Equation, and it would be what one could call a linear combination of a and b, such that for some k and j:

And if we say that d is the gcd(a,b), we can see that if d does not divide c, the equation cannot hold (if they are integers of curse). In such a way that for some l such that c=ld:

And so we have found a solution for the Linear Diophantine Equation; namely, x=x₀=lk and y=y₀=lj. But what about other solutions? Since, as I have spoken before,, the solutions for this kind of equation lie on a straight line, so what does that say? Well, to understand we must first consider another solution for this equation, call them (x’,y’), such that:

Now solving for x’ and y’, we can see that clearly:

But that will mean that all solutions can be found after selecting some t and doing the transformations shown above. Therefore all solutions for a Linear Diophantine Equation can be found just from one solutions, it does not necessarily work only for the linear combination for the gcd(a,b). Such that we can select an example, a=6, b=8 and c=2, we can see clearly that x₀=-1 and y₀=1, so we can see that the other solutions for this Linear Diophantine Equation are going to be among x₀=…,11,7,3,-1,-5,-9,-13,… y₀=…,-8,-5,-2,1,4,7,10,…

And thus we have found. This proof I have shown I read on a very famous book of Number Theory of the Dover Publications called “Number Theory”.

Soli Deo Gloria

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