How to know an equilateral hyperbola equation has or don't has integer solution?

Question

This is the equation

$$y=(K-3x)/(1+2x)$$

$K$ is a positive integer

I want to know for any given $K>0$, without plotting a graph, is it possible to know this equation has positive integer solutions or not?

It may be simple but I already left school for decades. I just come across maths problem again and I find it interesting. So please help.

Many Thanks in advance.

Answer 1

Suppose there is a positive integer solution, that is, for some positive integers $x,y$:

$$K - 3x = y(1+2x)$$

Observe that: $$2K + 3 = (2K-6x)+(3+6x) = (2y+3)(1+2x)$$

which is also a multiple of $1+2x$.

So we only need to check whether $2K+3$ has factors of the form $1+2x$.

But $2K+3$ is odd, so all its factors are odd.

However we require $x, y > 0$. This leads to (edited): $$x< \frac K3 \implies 3 \le 1+2x < 1+\frac {2K}3 = \frac {2K+3}3$$

and we can show that, whenever $2K+3$ is not prime and greater than $9$, there is an odd factor of $2K+3$ satisfying the above inequality.

Therefore there is a positive integer solution whenever $2K+3$ is composite and $>9$.

Answer 2

$$ (2x+1)(2y+3) = 4xy + 6x + 2y + 3 $$

You have $$ 4xy + 6x + 2y = 2K $$

$$ (2x+1)(2y+3) = 3 + 2 K $$

The integer solutions come from factoring $3+2K,$ making a list of all divisors (positive and negative), assigning one divisor to $(2x+1)$ and the appropriate quotient to $(2y+3)$