<s>[INST] Prove that there are no such positive integers $a,b,c,d$ such that $a^2 + b^2 = 3(c^2 + d^2)$ Prove that there are no positive integers a, b, c, d such that $a^2 + b^2 = 3(c^2 + d^2)$. Hint: What can you say about divisibility of a and b by 3? Look at solution with smallest possible a. 
 [/INST] Well, $A^2+B^2=1,0$ ( mod 4) so it couldn't be $A^2+B^2=4k+3$. But (4k+1)$(4k+3)=16k^2+16k+3=4(4k^2+4k)+3=4r+3$, while (4k+3)$(4k+3)=16k^2+24k+8+1=4(4k^2+6k+2)+1=4s+1$. So if a prime $p|A^2+B^2$ and p=4k+3, $p^2|A^2+B^2!$ But the exponent of number 3 is odd while it should be even. Hence there no positive integers A,B,C,D such that $A^2+B^2=3(C^2+D^2)$.
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