Let $f\_1,f\_2,f\_3,f\_4,f\_5 \in \mathbb{Q}[x]$ be linear and
coprime and not all constant.



> 
> Is it possible $ f\_1^2+f\_2^2+f\_3^2+f\_4^2=f\_5^2$?
> 
> 
> 


I suppose the answer is negative.


If this is possible, solving $f\_5(x)=N$ would give
deterministic representation of $N$ as sum of four
squares (probabilistic algorithms exist).


Couldn't solve this by equating coefficients.



> 
> What is the smallest natural $k$ such that
>  subject to the same constraints $f\_1^2+f\_2^2+\cdots+f\_k^2=f\_{k+1}^2$?
> 
> 
> 



Would such polynomial identity exist? (related to sum of four squares)
To see that $f\_{1}$, $f\_{2}$, $\ldots$, $f\_{k}$ must be scalar multiples of $f\_{k+1}$, plug in the root of $f\_{k+1}$ into both sides of the equation.


