Problem Statement:
Question-1:
For every natural number $n \geq 1$ , show that there exists $(a_1,...,a_n) \in \mathbb{N}^*$ s.t :
$a_1^2+a_2^2+...+a_n^2$ is a perfect square.
Question-2:
Show that there exists a sequence $(a_n)_{n\geq 1} \in \mathbb{N}$ s.t :
$\forall n \in \mathbb{N} : a_1^2+a_2^2+..+a_n^2$ is a perfect square .
My approach:
I tried Using principle of mathematical induction to solve the first Question :
for $n=1$ , it suffices to choose $a_1=m$ with $m\in \mathbb{N}$
for $n=2$ , any pyhtagorean triple will work , we can choose for instance $a_1=3$ and $a_2=4$ , it will yield $a_1^2+a_2^2=5^2$
Let $n \geq 2$ such that :$ \exists (a_1,...a_n) \in \mathbb{N}$ s.t $S_n=a_1^2+...+a_n^2$ is a perfect square (i.e $ S_n=k^2 $ with $k\in \mathbb{N}$)
To prove the statement is true for $n+1$ it suffices to show that :
$$\exists (a,z)\in \mathbb{N}^2 : a^2+k^2=z^2$$ and then set $a_{n+1}=a$
I get stuck here in fact i don't think a solution exists for all k.
Any hint would be greatly appreciated.