I'm quite new to number theory and I'm studying diophantine equations. I noticed that the technique was used for solving integer solutions. However, what technique can I use for solving natural number solutions?
For example, $ax + by = c$ find natural number solutions for $a$ and $b$
The only guess I can come up with is to set x and y to be absolute value:
$a|x| + b|y| = c$
But I'm not really sure how to solve this. Is there any other way I can go about finding only natural number solutions? Any help would be appreciated, thank you!
Euclidean algorithm guarantees existence of $a_0x+b_0y=c$. I am assuming you know how to find one solution. Assuming $\gcd(x,y)=1$, I think all solutions are given by
$$ a = a_0 + kb_0y \\ b = b_0 - ka_0x $$
I'm not sure how to prove this, though. But if you want to only find all natural number solutions, you just need to find values of $k$ such that both $a$ and $b$ are positive.
[Claim] If $\gcd(x,y)=d$ then all solutions are given by $$ a = a_0 + k \frac{b_0}{d} y \\ b = b_0 - k \frac{a_0}{d} x $$