Are there any positive odd integer solutions to $c^2=a^2-b^2+ab$

Question

I got to this point after trying to solve a special case of the rational distance problem.

The general problem is as follows:

Consider the points $(0,0), (0,1), (1,0), (1,1)$. Is there a point on the real plane where the distance to each point on that square is rational?

I was trying to prove that there is no solution when the point in question lies on the x axis.

After a lot of rearranging, I ended up with the diophantine equation, $c^2=a^2-b^2+ab$ where $a,b,c$ are all relatively prime to one another, and $a,b$ are odd?

My gut instinct tells me that there aren’t any nontrivial solutions, but I cant figure out how to prove it.

More generally, is there a way to determine if $c^2=xa^2+yab+zb^2$ has solutions in the positive integers, given $x,y,$ and $z$?

Edit: $a$ and $b$ are both odd.

Answer 1

$$ \left( x^2 + xy - y^2 \right)^2 = X^2 + XY - Y^2 $$

where $$ X = x^2 + y^2 $$ $$Y = 2xy + y^2 $$

Answer 2

Let $x=a/c$ and $y=b/c$. All rational points on $x^2 - y^2 + xy = 1$ are solutions. Since $(-1, 0)$ is a known solution, consider all lines passing through that point with a slope of $m/n$: $y=(x+1)m/n$. (https://www.desmos.com/calculator/kcrsllei9b) For the intersection of that line and the hyperbola, if $(m,n)$ are integers, then $(x,y)$ must be rational.

Solving the intersection of the line and parabola gives

$$x=\frac{n^2+m^2}{n^2+mn-m^2}$$ $$y=\frac{2mn+m^2}{n^2+mn-m^2}$$

This gives all (primitive) solutions as $$a = n^2 + m^2$$ $$b = 2mn + m^2$$ $$c = n^2 + mn - m^2$$

For example, if $m=1$ and $n=7$ then $(a,b,c) = (10,3,11)$.

---

If you are only interested in odd solutions, you can look at the 4 cases of $m$, $n$ even or odd. Or more directly, $a = n^2 + m^2$ requires $m,n$ to have opposite parity, $b = 2mn + m^2$ forces $m$ odd and $n$ even, which forces $c = n^2 + mn - m^2$ to make $c$ odd.

So the odd solutions come from $m$ odd and $n$ even, such as $(m,n)=(1,4)$ yields $(a,b,c) = (17,9,19)$.

Answer 3

$$\{a \to 240, b \to 72, c \to 264\}, \{a \to 338, b \to 48, c \to 358 \}, \{ a \to 73, b \to 57, c \to 79 \}$$