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.


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.


$i-$ 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…..


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?

Solutions Collecting From Web of "Representation of a number as a sum of squares."

It turned out the following. For a square shape


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.


The solution is easy to write.







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.