<s>[INST] Do four natural numbers exist which satisfy these constraints? Do four natural numbers $a,b,c$ and $d$ exist such that the following three conditions are true?
$$a^2+b^2+2d^2=c^2$$
$$\sqrt{a^2+d^2}\in\mathbb{N}$$
$$\sqrt{b^2+d^2}\in \mathbb{N}$$
 [/INST] You want $(a^2+d^2)+(b^2+d^2)=c^2$ and also $a^2+d^2=x^2$ and $b^2+d^2=y^2$.
The solutions of $x^2+y^2=c^2$ are $x=2mnk,y=(m^2-n^2)k,c=(m^2+n^2)k$ where $(m,n)=1$ and $m>n$.
Now we have the equations $a^2+d^2=(2mnk)^2$ and $b^2+d^2=((m^2-n^2)k)^2$.
Do the same and see what happens:)
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