I am asked to show that there are no non zero integer solutions to the following equation $x^2=3y^2+3z^2$
I think that maybe infinite descents is the key.
So I started taking the right hand side modulo 3 which gives me zero. Meaning that $X^2$ must be o modulo 3 as well and so I can write $X^2=3k$ , for some integer K and (k,3)=1.
I then divided by 3 and I am now left with $k=y^2+z^2$ . Now I know that any integer can be written as sum of 2 squares if and only if each prime of it's prime factorization has an even power if it is of the form 4k+3. But yet I am stuck . If anyone can help would be appreciated.