Integer as Sum of Three Squares

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Theorem

Let $r$ be a positive integer.

Then $r$ can be expressed as the sum of three squares if and only if it is not of the form:

$4^n \paren {8 m + 7}$

for some $m, n \in \Z_{\ge 0}$.


Sequence

The sequence of positive integers that cannot be expressed as the sum of at most $3$ squares begins:

$7, 15, 23, 28, 31, 39, 47, 55, 60, \ldots$


Proof

Sufficient Condition

Suppose $r$ is not of the form $4^n \paren {8 m + 7}$.

Then we need to show that it can always be expressed as the sum of three squares.



Necessary Condition

From Square Modulo 8, the squares modulo $8$ are $0, 1$ and $4$.

So the sum of three squares can be congruent modulo $8$ to any of the values $0, 1, 2, 3, 4, 5$ or $6$, but not $7$.

So no number of the form $8 m + 7$ can be the sum of three squares.


Now, suppose that $\exists n \ge 1, m \ge 0$ such that:

$4^n \paren {8 m + 7} = x^2 + y^2 + z^2$

As the left hand side is congruent modulo 4 to $0$, and as squares modulo 4 are either $0$ or $1$, it must be that $x, y$ and $z$ are all even.

Putting $x = 2 x_1, y = 2 y_1, z = 2 z_1$, we get:

$4^{n - 1} \paren {8 m + 7} = x_1^2 + y_1^2 + z_1^2$

If $n - 1 > 1$ then $x_1, y_1$ and $z_1$ are all still even, and the argument can be repeated:

$4^{n - 2} \paren {8 m + 7} = x_2^2 + y_2^2 + z_2^2$

Thus we descend through all powers of $4$ till $8 m + 7$ itself is expressed as the sum of three squares.

But this is impossible, as we saw above.

So the assumption that $4^n \paren {8 m + 7}$ can be expressed as the sum of three squares is false.

$\blacksquare$


Sources