Non-linear Diophantine equation on integer quadruples

Find all integer quadruples $\{a,b,c,d\}$ such that

$$ad = b + c$$

$$bc = a^2 - d$$

Working $\bmod 8$ (very messy) gives $d = 3 - 8k \quad \forall k \in \mathbb{N}$.

Numerical searching has so far only found $d = -5$ works.

• You did something wrong when you worked mod $8$: $(-10, -1000, 0, 100)$ is a solution, and Maple found lots of them using brute force (there are $59$ solutions where $|a|, |c| \le 10$). – Christopher Carl Heckman Apr 5 '16 at 6:51
• Actually, there's infinitely many integer solutions with $d=-5$. Let, $a,\,b,\,c = -x-5y,\;2y,\;5x+23y$ and $x,y$ solve the Pell-like equation $$x^2-21y^2=-5$$ From the initial solution $x,\,y = 4,\,1$ you get an infinite more. – Tito Piezas III Apr 10 '16 at 5:20
• With another $d$, you'll still get a similar Pell-like equation. – Tito Piezas III Apr 10 '16 at 5:24

For the system. \left\{\begin{aligned}&ad=b+c\\&bc=a^2-d\end{aligned}\right.

We reduce to one equation.

$$c=ad-b$$

It is necessary to solve the General Pell equation.

$$a^2-dab+b^2=d$$

You can make the change. $a\longrightarrow{a+tb}$

$$a^2+(2t-d)ab+(t^2+1)b^2=d$$

Then you can use this formula. And to reduce to writing the solution in the equation Pell. It is only necessary that the root was intact at least. http://www.artofproblemsolving.com/community/c3046h1048219 It is only necessary that the root was intact at least. $$\sqrt{\frac{d}{(t+1)^2+1-d}}=k$$