Is there any solution to the following system of equations? Is there any solution to the following system of diophantine equations?
$$
\left\{\begin{array}{l} 
2.a^2 = b^2+c^2+d^2 \\
a^2 = e^2+f^2+g^2 , & \mbox{with }((a,b,c,d,e,f,g)>2)\in N\mbox{ and differents among them}
\end{array}
\right.
$$
 A: Yes there are solutions.  For your first, note that if $a,b,c$ is a Pythagorean triangle with $a$ the hypotenuse and $a=d$ you have a solution.  There are more.  Since the equations are homogeneous (all the terms are squared) any multiple of a solution is another solution.  The easy one is then $a=15,b=9,c=12,d=15,e=f=10,g=5$ Maybe the common values violate "differents among them" but I don't know how to read that.  But then you can use Oleg567's solution and scale up.
A: Let $c=d=e$, and let $f=g$.  This gives us the following system of equations:
$2 \left( a^2-c^2\right)=b^2$
$a^2-c^2=2f^2$
If you further make the substitution $c=a-2$ and $b=2f$, we can combine both equations to yield one equation:
$2 \left( a-1\right)=f^2$
A: In the system of equations:
$\left\{\begin{aligned}&X^2+Y^2+Z^2=2R^2\\&F^2+G^2+T^2=R^2\end{aligned}\right.$
Solutions have the form:
$X=4(c^2+f^2+k^2)k^2-(2c^2-2d^2+f^2)f^2-(d^2-c^2)^2$
$Y=4kd(2k^2+d^2-c^2-f^2)$
$Z=4(f^2+c^2-k^2)k^2+(f^2+2c^2-2d^2)f^2+(d^2-c^2)^2$
$F=4kc(2k^2+d^2-c^2-f^2)$
$G=4kf(2k^2+d^2-c^2-f^2)$
$T=(2k^2+d^2)^2+(c^2+f^2)^2-2(d^2+4k^2)(c^2+f^2)$
$R=4(k^2+d^2)k^2+(2c^2+f^2-2d^2)f^2+(d^2-c^2)^2$
$k,c,d,f$ - integers and sets us.
A: set of equations:
$\left\{\begin{aligned}&X^2+Y^2+Z^2=2(R^2+W^2)\\&F^2+G^2+T^2=R^2+W^2\end{aligned}\right.$
has solutions
$X=4(t^2+q^2-a^2-v^2)a^2+(v^2+t^2+q^2-z^2)^2$
$Y=4(v^2-a^2-t^2-q^2)a^2+(v^2+t^2+q^2-z^2)^2$
$Z=4az(v^2+t^2+q^2-2a^2-z^2)$
$F=4(a^2+z^2-2t^2-2q^2)a^2+(v^2+t^2+q^2-z^2)^2$
$G=4aq(v^2+t^2+q^2-2a^2-z^2)$
$T=4at(v^2+t^2+q^2-2a^2-z^2)$
$R=4(z^2+a^2-2v^2)a^2+(v^2+t^2+q^2-z^2)^2$
$W=4av(v^2+t^2+q^2-2a^2-z^2)$
$a,v,t,q,z$ - integers and sets us.
