Download Representing Numbers as the Sum of Two Squares

Representing Numbers as the Sum
of Two Squares
Chris Peikert
Representing Numbers as the Sum of Two Squares – p.1/13
Question: Which numbers can be
written as the sum of two squares?
Representing Numbers as the Sum of Two Squares – p.2/13
Answer: Exactly those numbers
whose prime divisors of the form
divide an even number of
times.
Representing Numbers as the Sum of Two Squares – p.3/13
Properties of
Agenda
Representing Numbers as the Sum of Two Squares – p.4/13
Numbers like
Properties of
Agenda
are bad
Representing Numbers as the Sum of Two Squares – p.4/13
Agenda
Properties of
Primes like
Numbers like
are bad
are good
Representing Numbers as the Sum of Two Squares – p.4/13
Agenda
Properties of
Primes like
Primes like
Numbers like
are bad
are good
are good
Representing Numbers as the Sum of Two Squares – p.4/13
Agenda
Properties of
Primes like
Primes like
Numbers like
are bad
are good
are good
Extend to all naturals
Representing Numbers as the Sum of Two Squares – p.4/13
Agenda
Properties of
Primes like
Primes like
Numbers like
are bad
are good
are good
Extend to all naturals
Face the consequences
Representing Numbers as the Sum of Two Squares – p.4/13
Crash Course in
is prime. Then:
Assume
are elements “mod ”
Representing Numbers as the Sum of Two Squares – p.5/13
Crash Course in
and
are elements “mod ”
;
is prime. Then:
Assume
Representing Numbers as the Sum of Two Squares – p.5/13
Crash Course in
are elements “mod ”
such that
and
;
is prime. Then:
Assume
Representing Numbers as the Sum of Two Squares – p.5/13
Crash Course in
are elements “mod ”
are all distinct for
such that
The elements
.
and
;
is prime. Then:
Assume
Representing Numbers as the Sum of Two Squares – p.5/13
Who needs imaginaries?
Primes fall into 3 categories. How many solutions
to
?
: 1 solution
Representing Numbers as the Sum of Two Squares – p.6/13
Who needs imaginaries?
: 1 solution
Primes fall into 3 categories. How many solutions
to
?
: 2 solutions
Representing Numbers as the Sum of Two Squares – p.6/13
Who needs imaginaries?
: 1 solution
: 2 solutions
Primes fall into 3 categories. How many solutions
to
?
: 0 solutions
Representing Numbers as the Sum of Two Squares – p.6/13
Even numbers:
is not representable
Representing Numbers as the Sum of Two Squares – p.7/13
Even numbers:
is not representable
Odd numbers:
Representing Numbers as the Sum of Two Squares – p.7/13
Even numbers:
is not representable
or
Thus,
Odd numbers:
Representing Numbers as the Sum of Two Squares – p.7/13
is representable
with
such pairs.
. There
Consider
are
Representing Numbers as the Sum of Two Squares – p.8/13
is representable
, there exist distinct
such that
. There
For any
with
such pairs.
Consider
are
.
Representing Numbers as the Sum of Two Squares – p.8/13
is representable
. There
.
, then
Let
and
, there exist distinct
such that
For any
with
such pairs.
Consider
are
.
Representing Numbers as the Sum of Two Squares – p.8/13
is representable
. There
and
:
, so
.
Let
.
, then
Let
and
, there exist distinct
such that
For any
with
such pairs.
Consider
are
Representing Numbers as the Sum of Two Squares – p.8/13
, part 1
Let’s prove it again. . .
Consider:
Representing Numbers as the Sum of Two Squares – p.9/13
, part 1
Let’s prove it again. . .
Map
Consider:
Representing Numbers as the Sum of Two Squares – p.9/13
, part 1
Let’s prove it again. . .
Map
Consider:
Define:
Representing Numbers as the Sum of Two Squares – p.9/13
, part 1
Let’s prove it again. . .
Map
Consider:
,
Since
Define:
, we get
.
Representing Numbers as the Sum of Two Squares – p.9/13
, part 2
Map
Representing Numbers as the Sum of Two Squares – p.10/13
, part 2
Check:
Map
is well-defined, and an involution
Representing Numbers as the Sum of Two Squares – p.10/13
, part 2
Check:
Map
is well-defined, and an involution
, so
and
which implies
Find ’s fixed point(s?):
.
Representing Numbers as the Sum of Two Squares – p.10/13
, part 2
Check:
Map
is well-defined, and an involution
, so
Conclude:
and
which implies
Find ’s fixed point(s?):
.
is odd.
Representing Numbers as the Sum of Two Squares – p.10/13
, part 3 (whew!)
Map
Representing Numbers as the Sum of Two Squares – p.11/13
, part 3 (whew!)
has a fixed point, i.e.:
is odd, so
Map
Representing Numbers as the Sum of Two Squares – p.11/13
, part 3 (whew!)
has a fixed point, i.e.:
is odd, so
Map
QED! (Again!)
Representing Numbers as the Sum of Two Squares – p.11/13
Extend to all naturals
1 and 2 are representable; so are primes
.
Representing Numbers as the Sum of Two Squares – p.12/13
Extend to all naturals
,
Closure under multiplication:
1 and 2 are representable; so are primes
.
.
Representing Numbers as the Sum of Two Squares – p.12/13
Extend to all naturals
1 and 2 are representable; so are primes
.
,
Closure under multiplication:
.
Multiplication by a square:
.
Representing Numbers as the Sum of Two Squares – p.12/13
Extend to all naturals
1 and 2 are representable; so are primes
.
,
Closure under multiplication:
.
Multiplication by a square:
.
If
divides
, then
divides and
divides , and
representable.
is
Representing Numbers as the Sum of Two Squares – p.12/13
Infinitude of Primes, Redux
Euclid says: “there are infinitely many primes
.”
Representing Numbers as the Sum of Two Squares – p.13/13
Infinitude of Primes, Redux
Euclid says: “there are infinitely many primes
.”
.”
Fermat says: “ditto for primes
Representing Numbers as the Sum of Two Squares – p.13/13
Infinitude of Primes, Redux
Euclid says: “there are infinitely many primes
.”
.”
Fermat says: “ditto for primes
Dirichlet says: “ditto for primes
, for
any co-prime
.” Proof doesn’t fit in the margin. . .
Representing Numbers as the Sum of Two Squares – p.13/13