Representation of a number as a sum of squares.

There was a discussion on this representation. Determining the number of ways a number can be written as sum of three squares

I was interested in a curious fact. To solve this equation.

$$N=x_1^2+x_2^2+x_3^2=x_4^2+x_5^2+x_6^2=x_7^2+x_8^2+x_9^2=x_{10}^2+x_{11}^2+x_{12}^2$$

There is a number, when $3$ of the square are $4$ ways. Formula parameterization when he looked it turned out that it is possible to find a number that 3 squares can be present many times.

When he looked at the sum of two squares, there are also.

$$N=x_{1,1}^2+x_{1,2}^2=x_{2,1}^2+x_{2,2}^2=....=x_{i,1}^2+x_{i,2}^2$$

$i-$ the number of options can be indefinitely large. You can always find a number that will be the solution.

This is role $2, 3,$ or more terms. In a more General view.....

$$N=x_{1,1}^2+x_{1,2}^2+...+x_{1,j}^2=....=x_{i,1}^2+x_{i,2}^2+...+x_{i,j}^2$$

For any given $i,j - $ you can always find the number and General infinitely many solutions.

The question is. Only such it is possible to obtain solutions with arbitrarily large number $i$? Or maybe there are other forms in which the same observed?

Solution 1:

It turned out the following. For a square shape

$$N=a_1x_{1,1}^2+a_2x_{1,2}^2+...+a_{j}x_{1,j}^2=......=a_1x_{i,1}^2+a_2x_{i,2}^2+a_{j}x_{i,j}^2$$

If any number of options $i,j$ and any coefficients $a_{j}$. Solutions is always there.

Illustrate $2$ coefficients. Similarly solved if the number of different factors. That was evident symmetry and do not get confused is better to take $3$ equation.

$$ax_1^2+bx_2^2=ax_3^2+bx_4^2=ax_5^2+bx_6^2$$

The solution is easy to write.

$$x_1=t(bp^2+ak^2)(bs^2+ak^2)$$

$$x_2=y(bp^2+ak^2)(bs^2+ak^2)$$

$$x_3=((bp^2-ak^2)t-2bpky)(bs^2+ak^2)$$

$$x_4=((bp^2-ak^2)y+2apkt)(bs^2+ak^2)$$

$$x_5=((bs^2-ak^2)t-2bsky)(bp^2+ak^2)$$

$$x_6=((bs^2-ak^2)y+2askt)(bp^2+ak^2)$$

Here the representation of 3 options, but it is easy to see that can be written in the form of a combination with any number of options.