<s>[INST] How to prove that the equation $3(a^2+b^2)=7(c^2+d^2)$ has no solution 
Question 1: Let $a,b,c,d$ be positive integers, show that
$$3(a^2+b^2)=7(c^2+d^2)$$ has no solution.

question 1 is from Mathlove, Curious How prove it?

My question 2: Find the least positive integer $k$ such that
$$a^2+b^2=k(c^2+d^2)$$ has no solution.

 [/INST] $$\frac{a^2+b^2}{c^2+d^2}=\frac{(a^2+b^2)(c^2+d^2)}{(c^2+d^2)^2}$$
Use Brahmagupta–Fibonacci identity for the numerator to get $$\frac{a^2+b^2}{c^2+d^2}=\left(\frac{ac\pm bd}{c^2+d^2}\right)^2+\left(\frac{ad\mp bc}{c^2+d^2}\right)^2$$
But $3,7\equiv-1\pmod4$  can not be expressed as the sum of two squares 
</s>