<s>[INST] Solving the Diophantine Equation $ax^2 + bx + c = dy^2 + ey + f$? As a fun topic for personal research, I am trying to determine everything I can about Pythagorean triples for which all three integers are one less than a perfect square (please don't say anything about this if there is a known solution, I want to try to figure it out myself).
Here is my work so far;
Let $a^2 + b^2 = c^2$ be a primitive triple (that is, $a,b,c$ are pairwise relatively prime). Then the objective is to find a positive integer $g$ so that for some positive integers $x,y,z$;
$$ag = x^2 - 1$$
$$bg = y^2 - 1$$
$$cg = z^2 - 1$$
This is equivalent to solving $\dfrac{x^2-1}{a} = \dfrac{y^2-1}{b} = \dfrac{z^2-1}{c} \in \mathbb{Z}^+$.
Now, consider the problem of solving for $x$. The equation given for $x$ can be written $x^2 \equiv 1 \pmod{a}$. Let $\alpha$ be a solution to this equation in the reduced residue system mod $a$. Then we have a family of solutions to the Diophantine equation $ag = x^2 - 1$ given by $x = at + \alpha, t \in \mathbb{N}$. This leads to
$$\dfrac{x^2-1}{a} = \dfrac{(at+\alpha)^2-1}{a} = at^2 + 2\alpha t + \dfrac{\alpha^2-1}{a}$$
Now, doing the same thing for $y$ and $z$, we can reduce the problem to finding $t,m,n$ such that
$$at^2 + 2 \alpha t + \dfrac{\alpha^2-1}{a} = bm^2 + 2 \beta m + \dfrac{\beta^2-1}{b} = cn^2 + 2 \gamma n + \dfrac{\gamma^2-1}{c}$$
In case it helps, I found a pretty nice way to factorize these
$$at^2 + 2 \alpha t + \dfrac{\alpha^2-1}{a} = a \bigg{(} t + \dfrac{\alpha+1}{a} \bigg{)} \bigg{(} t + \dfrac{\alpha-1}{a} \bigg{)}$$
with the equivalent factorizations holding for the other expressions as well.
where $\alpha^2 \equiv 1 \pmod{a}, \beta^2 \equiv 1 \pmod{b}, \gamma^2 \equiv 1 \pmod{c}$. So my question is whether there is any method for solving an equation like this, even solving the equations pairwise, in the general case or for any special properties of $a,b,c,\alpha,\beta,\gamma$.
Here are all the solutions of which I am aware;
When $a,b,c = 33,56,65$, we can set $x,y,z = 10,13,14$. This yields the equation $99^2 + 168^2 = 195^2 \implies (3a)^2 + (3b)^2 = (3c)^2$. 
When $a,b,c = 133,156,205$, we have $x,y,z = 265, 287, 329$. This yields the equation $70224^2 + 82368^2 = 108240^2 \implies (528a)^2 + (528b)^2 = (528c)^2$.
Thanks in advance!
 [/INST] Legendre gave a general solution to the equation
$$
ax^2+bxy+cy^2+dx+ey+f=0.
$$
Your equation/system can certainly be put in this form, and then the parameterization given by Legendre would provide you with the answers you seek.
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