Question

A^2+B^2+C^2+D^2 = N Given an integer N, print out all possible combinations of integer values of ABCD which solve the equation.

I am guessing we can do better than brute force.

Answer 1

The Wikipedia page has some interesting background information, but Lagrange’s four-square theorem (or, more correctly, Bachet’s Theorem – Lagrange only proved it) doesn’t really go into detail on how to find said squares.

As I said in my comment, the solution is going to be nontrivial. This paper discusses the solvability of four-square sums. The paper alleges that:

There is no convenient algorithm (beyond the simple one mentioned in
the second paragraph of this paper) for finding additional solutions
that are indicated by the calculation of representations, but perhaps
this will streamline the search by giving an idea of what kinds of
solutions do and do not exist.

There are a few other interesting facts related to this topic. There
exist other theorems that state that every integer can be written as a
sum of four particular multiples of squares. For example, every
integer can be written as N = a^2 + 2b^2 + 4c^2 + 14d^2. There are 54
cases like this that are true for all integers, and Ramanujan provided
the complete list in the year 1917.

For more information, see Modular Forms. This is not easy to understand unless you have some background in number theory. If you could generalize Ramanujan’s 54 forms, you may have an easier time with this. With that said, in the first paper I cite, there is a small snippet which discusses an algorithm that may find every solution (even though I find it a bit hard to follow):

For example, it was reported in 1911 that the calculator Gottfried
Ruckle was asked to reduce N = 15663 as a sum of four squares. He
produced a solution of 125^2 + 6^2 + 1^2 + 1^2 in 8 seconds, followed
immediately by 125^2 + 5^2 + 3^2 + 2^2. A more difficult problem
(reflected by a first term that is farther from the original number,
with correspondingly larger later terms) took 56 seconds: 11399 = 105^2
+ 15^2 + 8^2 + 5^2. In general, the strategy is to begin by setting the first term to be the largest square below N and try to represent the
smaller remainder as a sum of three squares. Then the first term is
set to the next largest square below N, and so forth. Over time a
lightning calculator would become familiar with expressing small
numbers as sums of squares, which would speed up the process.

(Emphasis mine.)

The algorithm is described as being recursive, but it could easily be implemented iteratively.