Download pythagorean_triples

© T Madas
Experience on using the Pythagoras Theorem
tells us that:
There are a few integer lengths for a triangle which
satisfy the Pythagorean law.
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We all know the famous “three” numbers 3,4,5
Some of us are aware of 5,12,13
Even fewer of us know the 8,15,17 and the 7,24,25
What about 20,21,29?
How many of us know the 12,35,37?
or 9,40,41?
Does 2000, 999999, 1000001 also work?
Is there an infinite number of Pythagorean Triples?
Is there a way to generate such numbers?
© T Madas
a
2
c
b
2 m ,n Î ¥
2
2
2
2
ù
é
ù the identity:
We écan generate
such triples
using
2
+ éë2mn ù
º êm + n ú
êëm - n ú
û
û
ë
û m> n³ 1
a
c
b
ém 2 - n 2 ù2 + é2mn ù2 º ém 2 + n 2 ù2
êë
ú
êë
ú
û
û ë
û
where m , n Î ¥
and
m> n³ 1
© T Madas
a
2
b
c
2
m ,n Î ¥
ém - n
é
+ éë2mn ù
º êm + n ú
êë
ú
û
û
ë
û m> n³ 1
n
1
2
3
4
5
m
Notes
2
2ù
2
2
2ù
The condition m > n ≥ 1 is placed so
that a > 0
1
It should be obvious that we can
extend this table producing an infinite
number of Pythagorean triples
2
a=3
b=4
c=5
3
a=8
b=6
c = 10
a = 5 This “formula” does not produce unique
b = 12 triples for different values of m and n
c = 13
4
a = 15
b=8
c = 17
a = 12
b = 16
c = 20
a=7
b = 24
c = 25
5
a = 24
b = 10
c = 26
a = 21
b = 20
c = 29
a = 16
b = 30
c = 34
a=9
b = 40
c = 41
© T Madas
© T Madas
An unexpected way of generating Pythagorean triples
is by adding unit fractions whose denominators are
consecutive odd or even integers.
The numerator and denominator of the resulting
fraction will always give the two shorter sides of a
Pythagorean triple.
1+1= 4
1 3 3
Squaring and adding gives the 3, 4, 5 triple
1 + 1= 8
3 5 15
Squaring and adding gives the 8, 15, 17 triple
1 + 1 = 12 Squaring and adding gives the 12, 35, 37 triple
5 7 35
1 + 1 = 16
7 9 63
Squaring and adding gives the 16, 63, 65 triple
© T Madas
An unexpected way of generating Pythagorean triples
is by adding unit fractions whose denominators are
consecutive odd or even integers.
The numerator and denominator of the resulting
fraction will always give the two shorter sides of a
Pythagorean triple.
1+ 1= 3
2 4 4
Squaring and adding gives the 3, 4, 5 triple
1 + 1= 5
4 6 12
Squaring and adding gives the 5, 12, 13 triple
1+ 1 = 7
6 8 24
Squaring and adding gives the 7, 24, 25 triple
1+ 1 = 9
8 10 40
Squaring and adding gives the 9, 40, 41 triple
© T Madas
Why does this method generates Pythagorean triples?
1+
n
1 = 1´ (n + 2) + 1´ n
n + 2 + n = 2n + 2
=
n + 2 n ´ (n + 2) (n + 2)´ n
n (n + 2)
n 2 + 2n
Squaring and adding these quantities should produce
a perfect square, if this method is to work
(2n + 2)2 + (n 2 + 2n )2 = 4n 2 + 8n + 4 + n 4 + 4n 3 + 4n 2
= n 4 + 4n 3 + 8n 2 + 8n + 4
= (n 2 + 2n + 2)2
© T Madas
© T Madas