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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
and
so
and
. Thus
so, since
is a field we can divide by
and see that
Thus the quadratic residue symbol
equals
.
However,
Proof.
[Proof of Theorem
1.1]
Suppose that
is of the form
, that
(exactly
divides) with
odd, and that
. Letting
,
we have
with
and
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
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.
Since
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
.
Next: Computing and
Up: Sums of Two Squares
Previous: Sums of Two Squares
William A Stein
2001-10-31