Which Numbers are the Sum of Two Squares?
The main goal of today's lecture is to prove the following theorem.
Theorem 1.1 A number is a sum of two squares if and only if all prime factors of of the form have even exponent in the prime factorization of .
Before tackling a proof, we consider a few examples.
Example 1.2
- .
- is not a sum of two squares.
- is divisible by because is, but not by since is not, so is not a sum of two squares.
- is a sum of two squares.
- is a sum of two squares, since and is prime.
- is not a sum of two squares even though .
In preparation for the proof of Theorem 1.1, we recall a result that emerged when we analyzed how partial convergents of a continued fraction converge.
Lemma 1.3 If and , then there is a fraction in lowest terms such that and
Proof. Let be the continued fraction expansion of . As we saw in the proof of Theorem 2.3 in Lecture 18, for each
Since is always at least bigger than and , either there exists an such that , or the continued fraction expansion of is finite and is larger than the denominator of the rational number . In the first case,
so satisfies the conclusion of the lemma. In the second case, just let .
Definition 1.4 A representation is primitive if .
Lemma 1.5 If is divisible by a prime of the form , then has no primitive representations.
Proof. If has a primitive representation, , then
Thus the quadratic residue symbol equals . However,
and
so and . Thus so, since is a field we can divide by and see that
Proof. [Proof of Theorem 1.1] Suppose that is of the form , that (exactly divides) with odd, and that . Letting , we have
with and
so a product of two numbers that are sums of two squares is also a sum of two squares.1Also, the prime is a sum of two squares. It thus suffices to show that if is a prime of the form , then is a sum of two squares.
is a square modulo ; i.e., there exists such that . Taking in Lemma 1.3 we see that there are integers such that and
If we write
then
and
But , so
Thus .
Because is odd, , so Lemma 1.5 implies that , a contradiction.
Write where has no prime factors of the form . It suffices to show that is a sum of two squares. Also note that
Since
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