Pythagorean Triples and
Fermat’s Last Theorem
Shobha Bagai
Institute of Lifelong Learning
University of Delhi

PYTHAGOREAN TRIPLES AND FERMAT’S LAST THEOREM
Papyrus Berlin:
"You are told the area of a square of 100 square cubits is equal to
that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are
the sides of the two unknown squares?"
In modern terms we would express this as:
If
x
2
+
y
2
=
100
and
x
=
3
4
y
, what are
x
and
y
?
A modern solution in this form might be
3
4
y
"
#
$
%
&
’
2
+
y
2
=
100
(
9
16
y
2
+
y
2
=
100
(
25
16
y
2
=
100
(
y
2
=
100
)
16
25
=
64
(
y
=
8,
x
=
6.
The Berlin Papyrus 6619, commonly known as the
Berlin Papyrus
is an ancient
Egyptian papyrus document from the Middle Kingdom. This papyrus was found at the
ancient burial ground of Saqqara.
The problem stated in the Berlin Papyrus suggests some knowledge of what would
later be named the Pythagorean theorem.
Hieroglyphic transcription of fragment 1 and 3
from the Berlin Papyrus 6619.
[
grid/BERLIN_001.htm
]
The Hieratic fragments from the Berlin
papyrus 6619.
[
grid/BERLIN_001.htm
]

INTRODUCTION
Pythagoras was a mathematician born in Greece in about 570 BC.
He was interested in mathematics, science and philosophy. He is
known to most people because of the
Pythagoras Theorem
that
is about a property of all triangles with a right angle.
However,
h
2
=
a
2
+
b
2
is true only for right-angled triangles.
For acute-angled triangles
h
2
<
a
2
+
b
2
For obtuse-angled triangle
h
2
>
a
2
+
b
2
Note that in any triangle, the longest side
h
cannot be longer than the sum of the other
two sides. So
h < a + b
.
If the two shorter sides of a right-angled triangle are 2 cm and 3 cm, then the length of
the hypotenuse is given by
h
2
=
2
2
+
3
2
=
13
h
=
13
=
3.60555
In the example above, we chose two whole-number sides and found the longest side,
which was not a whole number.
Is it possible to construct a right triangle whose sides are whole numbers?
Many of us are familiar with right triangles of sides 3, 4, 5 or 5, 12, 13. These are
called the
Pythagorean triples
or
Pythagorean triads
.
If the three integers a, b, c have no common factor, then they are called
Basic
Pythagorean triples
. Thus, every Pythagorean triple is a basic or a multiple of a basic
triple. For e.g. the Pythagorean triple 9, 12, 15 is a multiple of the basic triple 3, 4, 5.
PYTHAGORAS THEOREM
If a triangle has one angle which is a right angle then there is
a special relationship between the lengths of its three sides
h
2
=
a
2
+
b
2
where
h
is the longest side called the hypotenuse and
a
and
b
are the other two sides.
Or
The square of the largest side is equal to the sum of the
squares of the other two sides in a right-angled triangle.
PYTHAGOREAN TRIPLE
Positive integers a, b, c that satisfy the relation
a
2
+
b
2
=
c
2
are called the Pythagorean triple.

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