Primitive Pythagoreans

The Pythagoreans were students of Pythagoras' school of thought.They subscribed to Pythagoreanism,a philosophy that was dominated by mathematics and mysticism.

A Pythagorean triple consists of three positive integers \((a,b,c)\) such that \(a^2+b^2=c^2\).

A triple is said to be primitive if \(a\),\(b\) and \(c\) are co-prime to each other( \(GCD(a,b,c)=1\) ).

There are \(16\) primitive triples with \(c \leq 100\) and \(a<b<c\) as shown below.

\(( 3, 4, 5 ),( 5, 12, 13),( 8, 15, 17),( 7, 24, 25)\)

\((20, 21, 29),(12, 35, 37),( 9, 40, 41),(28, 45, 53)\)

\((11, 60, 61),(16, 63, 65),(33, 56, 65),(48, 55, 73)\)

\((13, 84, 85),(36, 77, 85),(39, 80, 89),(65, 72, 97)\)

Find the number of primitive Pythagorean triples with \(c \leq 10^6\) and \( a < b < c \) ?

Details and assumptions

  • You may assume that the triplets \((a,b,c)\) and \((b,a,c)\) are the same when counting.
  • This was inspired by a projecteuler problem
  • For more about Pythagoreans and irrational numbers take a look at Peter's note on the discovery of irrational numbers.
  • There is a great interactive tutorial on pythagorean triples here.
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