<s>[INST] How to represent Fermat number $F_n$ as a sum of three squares? Let $F_n=2^{2^n}+1$  be  the Fermat number. How to represent the Fermat number  $F_n$ for $n \geq 3$  as a sum of three squares  of different natural numbers? For example  for   $n=3$  we have
$$
F_3=257=5^2+6^2+14^2.
$$
Is there any simple procedure  to write out such representations for another $n$?
 [/INST] Let $X=2^{2^{n-1}}$. Then $X$ is congruent to $1$ modulo $3$, so
$$
F_n=X^2+1=\bigg(\frac{2X+1}{3}\bigg)^2+
\bigg(\frac{2X-2}{3}\bigg)^2+
\bigg(\frac{X+2}{3}\bigg)^2
$$
The values are all distinct, except when $X=1$ or $4$ (which are excluded since
$n\geq 3$).
</s>