Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$ I'm something like 90% sure that this diophantine equation has nontrivial solutions:
$3(x^2+y^2+z^2)=10(xy+yz+zx)$
However, I have not been able to find a solution using my calculator. I would greatly appreciate if someone could try to find one using a program. Or maybe you can just guess one that happens to work?
Thanks!
EDIT: By nontrivial I mean no $0$'s. (Credits to Slade for reminding me to define this)
EDIT2: In fact, you are free to find a nontrivial solution to $(3n-3)(x^2+y^2+z^2)=(9n+1)(xy+yz+zx)$ where $n\equiv 1\pmod 5$ is a positive integer. The one I posted above is the case $n=5(2)+1$, but you will make my day if you can find a nontrivial solution for any $n=5k+1$.

As far as I understand - this is the site for solving the problem.  Programming and calculation using the computer is not mathematics.  If you want to calculate - there is a special section.   https://mathematica.stackexchange.com/questions
Here it is necessary to solve the equations.
For the equation:
$$3(x^2+y^2+z^2)=10(xy+xz+yz)$$
The solution is simple.
$$x=4ps$$
$$y=3p^2-10ps+7s^2$$
$$z=p^2-10ps+21s^2$$
$p,s - $ any integer which we ask.
Why make a program? What's the point? For what?
