Sum of Two Squares

Lemma.

If and are each the sum of two squares then so is there product

Proof.

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Theorem.

A prime of form can not be written as a sum of two squares

Proof.

□

Lemma.

let be a prime and let
The the congruenc

admits a solution where
and

Proof.

consider the set

Note that
Pigeonhole Principle Pigeonhole Principle implies that
where or such that

take,

Note:

□

Theorem.

An odd prime is expressed as a sum of two squares iff

Proof.

“only if” part is done in 2. Sum of Two Squares > ^8e7812 2. Sum of Two Squares > Theorem 2.2
let There exists an integer a such that

Please help me again
In fact, we can take where is the primitives root of
Note that
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Corollary (Uniqueness of

such that

Proof.

lets assume W.L.O.G

Then

similarly we can show

if we chose then we get and

hence They are unique
□

Theorem.

let the positive integers n be written as where is square-free. Then can be represented as the sum of two squares if and only if contains no prime factor of the form

Proof.
Suppose where has no prime factor of the form if then done.
Let write .
Each of the prime , being of the form , can be written as the sum of squares,
which implies there product also can be written as .

The other way.
Suppose,

assume that and let be an odd prime divisor of
let then
and where
This gives
since is square-free we have

since , cannot divide both and

we can find such that