The smallest dot product

Question

Consider an $n$-dimensional vector $V$ $(C_1 ,C_2 , . .. ,C_n)$ such that $C_i$ is a Whole number.
Is it possible to find a vector such that the dot product with itself is a perfect square ?
If possible find such vector , provided $ n $.

Answer 1

Every positive integer (and hence also every square) is the sum of $4$ squares. This is Lagrange's four-square theorem. Thus we have $$ m^2=x_1^2+x_2^2+x_3^2+x_4^2. $$

Note: the edited question asks for a solution of $$ m^2=x_1^2+\cdots +x_n^2, $$ where $1\le x_i\le 9$ are integers. In other words, given $n\ge 1$, find a sum with $n$ terms with each term in $\{1,4,9,16,25,36,49,64,81\}$, which gives a square.

Example: $n=5$. Then a solution is $1^2+1^2+1^2+2^2+3^2=4^2$. I suppose this has always a solution for every $n$. For the "minimal magnitude" one could think that one should use many summands $1^2$, but this is not clear. This could be a difficult problem.