Objectives
History of Fermat’s Last Theorem
Pythagorean Triples
MATH 135
Faculty of Mathematics, University of Waterloo
Lecture 19: Introduction to Fermat’s Last Theorem
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
History of Fermat’s Last Theorem
Pythagorean Triples
Content
Content
1.Provide an historical introduction.
2.Define gcd(x,y,z).
3.
State:
If x, y and z are integers, not all zero, and
gcd
(
x
,
y
) = 1
, then
gcd(
x
,
y
,
z
) = 1
.
4.Define aPythagorean tripleandprimitivePythagorean triple.
5.
State and prove:
Let d
= gcd(
x
,
y
,
z
)
. The three integers x, y
and z are a Pythagorean triple if and only if the three integers
x
1
=
x
/
d, y
1
=
y
/
d and z
1
=
z
/
d are a Pythagorean triple.
6.
State and prove:
If x, y and z are a primitive Pythagorean
triple, then x, y and z are relatively prime.
7.
State and prove:
If x, y and z are a primitive Pythagorean
triple, then one of the integers x or y is even and the other is
odd.
8.
State and prove:
If ab
=
c
n
and
gcd(
a
,
b
) = 1
, then there
exist integers a
1
and b
1
so that a
=
a
n
1
and b
=
b
n
1
.
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
History of Fermat’s Last Theorem
Pythagorean Triples
Diophantus’
Arithmetica
Pierre de Fermat (1601 (?) – 1635) was a brilliant French
mathematician. It was his habit to make notes in the margins of
his books and one such note is famous. Fermat possessed a copy
of Bachet’s translation of Diophantus’
Arithmetica
. Problem II.8
of the
Arithmetica
reads
Partition a given square into two squares.
Diophantus did not require the squares to be integers so we might
write Problem II.8 as
For what positive rational numbers
x
,
y
and
z
is the equation
x
2
+
y
2
=
x
2
satisfied?
Faculty of Mathematics, University of Waterloo
MATH 135

Objectives
History of Fermat’s Last Theorem
Pythagorean Triples
The Note
Adjacent to Problem II.8, and in the margin of his copy of
Arithmetica

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- Fall '08
- ANDREWCHILDS
- Math, Algebra, Pythagorean Theorem, Euclidean algorithm, pythagorean triples, Fermat's Last Theorem