This is from a new book I started reading; it asks whether I can construct another solution for $x^2=2y^2+7$, given that $(x\_0,y\_0)$ is one. It gives the following diagram as a hint:

[![enter image description here](https://i.stack.imgur.com/8ClAG.jpg)](https://i.stack.imgur.com/8ClAG.jpg)

However, this diagram only "tells me" that there are no integer solutions ($\neq 0$) for $x^2=2y^2$, and I really don't see how I can use that information. I wrote a little program to find the solutions up to 1000:

```
x | 3  5  13  27  75  157  437  915
y | 1  3   9  19  53  111  309  647

```

but cannot work out any patterns (except for all the numbers being odd). Any further hints would be appreciated.

**EDIT:** The table above is supposedly superfluous. I am really curious as to how the diagram constitutes a hint.

Here's the German excerpt:

> 
> IV . Die Basis eines gleichschenkeligen rechtwinkligen Dreiecks sei $x$ und die Schenkellänge sei $y$.
> 
> 
> a) [...]
> 
> 
> b) Entwickle aus der Zeichnung ein Verfahren, aus einer Lösung $(x,y)$ der Gleichung $x^2-2y^2=7$ eine weitere zu erhalten.
> 
> 
> c) [...]
> 
> 
> 

What might be relevant, is that in one of the previous examples it was shown, that if $(x\_0,y\_0)$ a soltion to $x^2+xy-y^2=1$, others can be obtained with $(y,x+y)$ and $(y-x,x)$, and that the same equation with a $0$ instead of a $1$ has no solutions.