Objectives
Fermat’s Last Theorem - Proof
Reducing FLT
History
MATH 135
Faculty of Mathematics, University of Waterloo
Lecture 21: Fermat’s Theorem for
n
= 4
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
Fermat’s Last Theorem - Proof
Reducing FLT
History
Content
Content
1.
State and prove:
The Diophantine equation x
4
+
y
4
=
z
2
has
no positive integer solution.
2.
State and prove:
The Diophantine equation x
4
+
y
4
=
z
4
has
no positive integer solution.
3.
Show reduction of FLT to
If p is an odd prime, then the
Diophantine equation x
p
+
y
p
=
z
p
has no positive integer
solution.
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
Fermat’s Last Theorem - Proof
Reducing FLT
History
1. Proof By Contradiction
2. Finding a Primitive Pythagorean Triple
3. What DosandtLook Like?
4. Finding Another Primitive Pythagorean Triple
5. The Contradiction
FLT, Strong Version of
n
= 4
Theorem (FLT, Strong Version of
n
= 4)
The Diophantine equation x
4
+
y
4
=
z
2
has no positive integer
solution.
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
Fermat’s Last Theorem - Proof
Reducing FLT
History
1. Proof By Contradiction
2. Finding a Primitive Pythagorean Triple
3. What DosandtLook Like?
4. Finding Another Primitive Pythagorean Triple
5. The Contradiction
Structure
1.
Proof by contradiction
2.
Finding a primitive Pythagorean triple
3.What dosandtlook like?
4.
Finding another primitive Pythagorean triple
5.
The contradiction
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
Fermat’s Last Theorem - Proof
Reducing FLT
History
1. Proof By Contradiction
2. Finding a Primitive Pythagorean Triple
3. What DosandtLook Like?
4. Finding Another Primitive Pythagorean Triple
5. The Contradiction
1. Proof By Contradiction
Suppose that there does exist a positive integer solution to
x
4
+
y
4
=
z
2
. Of all such solutions
x
0
,
y
0
,
z
0
, choose any one in
which
z
0
is smallest. Without loss of generality, we may also
assume that gcd(
x
0
,
y
0
) = 1. (Why?) This in turn implies that
gcd(
x
0
,
y
0
,
z
0
) = 1. (Why?)
Note: The last sentences of the proof are:
But recall that
x
0
,
y
0
,
z
0
is a solution to
x
4
+
y
4
=
z
2
with the
smallest possible value of
z
. But
x
1
,
y
1
,
z
1
is a solution to
x
4
1
+
y
4
1
=
z
2
1
with a smaller value of
z
!
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
Fermat’s Last Theorem - Proof
Reducing FLT
History
1. Proof By Contradiction
2. Finding a Primitive Pythagorean Triple

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- Fall '08
- ANDREWCHILDS
- Math, Algebra, Number Theory, Prime number, Fermat's Last Theorem, primitive Pythagorean triple