# Integers which are the sum of non-zero squares

Lagrange's four-square theorem states that every natural number can be written as the sum of four squares, allowing for zeros in the sum (e.g. $6=2^2+1^2+1^2+0^2$). Is there a similar result in which zeros are not allowed in the sum? For example, does there exist $n\in\mathbb{N}$ such that every natural number greater than $n$ can be written as the sum of five non-zero squares, or six non-zero squares, for example?

• Primes of the form $4k+1$ can be represented as $a^2+b^2$... Aug 26 '15 at 10:43
• Try this  Table[PowersRepresentations[m, 5, 2], {m, 1, 100}]  at W|A...looks like for $n>33$ you also always have five non-zero squares in your sum... Aug 26 '15 at 11:07

This question is answered in "Introduction to Number Theory" by Niven, Zuckerman& Montmogery (pp.318-319 of the fifth edition). I summarize their proof below.

Every integer $$\geq 34$$ is a sum of five positive squares (while $$33$$ is not). The number five is optimal, because the only representation of $$2^{2r+1}$$ as a sum of four squares is $$0^2+0^2+(2^r)^2+(2^r)^2$$ (easy exercice by induction on $$r$$).

One can check by hand by noting all the numbers between $$34$$ and $$169$$ are sums of five positive squares. Now, let $$n\geq 169$$ and let us show that $$n$$ is a sum of five positive squares.

We know that $$n-169$$ is a sum of four not necessarily positive squares, $$n-169=x_1^2+x_2^2+x_3^2+x_4^2$$ and we can assume $$x_1 \leq x_2 \leq x_3 \leq x_4$$.

If $$x_1>0$$, writing $$n=13^2+x_1^2+x_2^2+x_3^2+x_4^2$$ we are done. So assume $$x_1=0$$.

If $$x_2>0$$, writing $$n=5^2+12^2+x_2^2+x_3^2+x_4^2$$ we are done. So assume $$x_2=0$$.

If $$x_3>0$$, writing $$n=3^2+4^2+12^2+x_3^2+x_4^2$$ we are done. So assume $$x_3=0$$.

If $$x_4>0$$, writing $$n=2^2+4^2+7^2+12^2+x_4^2$$ we are done. So assume $$x_4=0$$.

So now all the $$x_i$$ are zero, and $$n=169=5^2+6^2+6^2+6^2+6^2$$. This concludes the proof.

• Is it straightforward, from here, to compute the number of ways that a number can be represented as a sum of five non-zero squares? In particular, I have been thinking of starting the question: how can one find the number, say $r_4^{*}(n)$, of ways to express a given integer $n$ as the sum of four non-zero squares? Feb 19 '18 at 10:19
• @user385459 I do not know. It does not seem straightforward to convert this existence proof into a counting method. Feb 19 '18 at 11:21