© T Madas

© 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.

• 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?

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2 222 2 2 22m n mn m né ù é ùé ù- + º +ê ú ê úë ûë û ë û

where , and 1m n m nÎ > ³¥

We can generate such triples using the identity:

a b c

2 222 2 2 22m n mn m né ù é ùé ù- + º +ê ú ê úë ûë û ë û 1m n> ³

a b c ,m n Î ¥

© T Madas

2 222 2 2 22m n mn m né ù é ùé ù- + º +ê ú ê úë ûë û ë û 1m n> ³

a b c ,m n Î ¥

1

a = 9

b = 40

c = 41

4

a = 16

b = 30

c = 34

a = 21

b = 20

c = 29

a = 24

b = 10

c = 265

a = 7

b = 24

c = 25

a = 12

b = 16

c = 20

a = 15

b = 8

c = 174

a = 5

b = 12

c = 13

a = 8

b = 6

c = 103

a = 3

b = 4

c = 52

5321m nNotes

The condition m > n ≥ 1 is placed so that a > 0

It should be obvious that we can extend this table producing an infinite number of Pythagorean triples

This “formula” does not produce unique triples for different values of m and n

© 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.11

13

+ 43

= Squaring and adding gives the 3, 4, 5 triple

13

15

+ 815

= Squaring and adding gives the 8, 15, 17 triple

15

17

+ 1235

= Squaring and adding gives the 12, 35, 37 triple

17

19

+ 1663

= Squaring and adding gives the 16, 63, 65 triple

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12

14

+ 34

= Squaring and adding gives the 3, 4, 5 triple

14

16

+ 512

= Squaring and adding gives the 5, 12, 13 triple

16

18

+ 724

= Squaring and adding gives the 7, 24, 25 triple

18

110

+ 940

= Squaring and adding gives the 9, 40, 41 triple

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.

© T Madas

2( )

Why does this method generates Pythagorean triples?1n

12n

++

1n= ( )

( )22

nn

´ +´ + ( )

12n

++

nn

´´ = 2n n+ +

( )2n n + 22 2

2n

n n+=

+

Squaring and adding these quantities should produce a perfect square, if this method is to work

2 2n + 2 22( )n n++ = 24n 8n+ 4+ 4n+ 34n+ 24n+

= 4n 34n+ 28n+ 8n+ 4+

2( )= 2n 2n+ 2+

© T Madas