Friday, 27 April 2012

Pythagorean triples

A Pythagorean triple (a,b,c) consists of positive integers a, b and c such that

a^2 + b^2 = c^2.\

The triple is called primitive if a, b and c are relatively prime.  It can be easily proved that all Pythagorean triples can be generated using Euclid's formula:

 a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2

where m and n are positive integers with m > n.  The triple (a,b,c) generated is primitive if m and n are coprime and m - n is odd.  There are 16 primitive Pythagorean triples with c < 100:

( 3 , 4 , 5 )( 5, 12, 13)( 7, 24, 25)( 8, 15, 17)
( 9, 40, 41)(11, 60, 61)(12, 35, 37)(13, 84, 85)
(16, 63, 65)(20, 21, 29)(28, 45, 53)(33, 56, 65)
(36, 77, 85)(39, 80, 89)(48, 55, 73)(65, 72, 97)


There are a lot of interesting facts about primitive Pythagorean triples.  For example:
  • One of a and b is odd, the other is even; c is always odd.
  • Exactly one of a and b is divisible by 3.
  • Exactly one of a and b is divisible by 4.
  • Exactly one of a, b and c is divisible by 5.
  • Every prime factor of c leaves a remainder of 1 when divided by 4.

A generalization of the Pythagorean triples leads to the famous Fermat's Last Theorem dated 1637, which remained an unsolved mystery for 358 years until finally proved by Andrew Wiles in 1995.  The theorem states that the equation

a^n + b^n=c^n \!

has no non-trivial solutions for (a,b,c) if n is a positive integer greater than 2.

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