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)(a,b,c) such that a2+b2=c2a^2+b^2=c^2.

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

There are 1616 primitive triples with c100c \leq 100 and a<b<ca<b<c as shown below.

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

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

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

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

Find the number of primitive Pythagorean triples with c106c \leq 10^6 and a<b<c a < b < c ?

Details and assumptions

  • You may assume that the triplets (a,b,c)(a,b,c) and (b,a,c)(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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