Pythagorean triple

Three positive integers that describe a right-angled triangle:

i ² + j ² = k ²

• There are an infinite number of Pythagorean triples.

  Euclid's proof : consider the identity n ² + 2 n + 1 = ( n +1) ²
  Whenever 2 n +1 is a square, this forms a Pythagorean triple. But 2 n +1 comprises all the odd numbers; every other square numbers is odd; there are an infinite number of odd squares; hence there are an infinite number of Pythagorean triples.
  

• Pythagorean triples constructed from Euclid's proof:

  [ m , where m = 2 k +1 ]	[ n = ( m 2 -1)/2 ]	[ n +1]
  3	4	5
  5	12	13
  7	24	25
  9	40	41
  11	60	61
  13	84	85
  15	112	113
  17	144	145
  19	180	181
  21	220	221	... etc ...

• There are an infinite number of Pythagorean triples not of this form, too. We can use the same trick on n ² + 4 n + 4 = ( n +2) ² . Whenever 4 n +4 = 4( n +1) is a square, we get a Pythagorean triple. Now we are looking for squares of all numbers divisible by 4, hence squares of even numbers. (However, if m is divisible by 2 but not by 4, if m =2(2 k +1), then the new triple is simply a multiple of a 'Euclidian' one. So we consider only multiples of 4.)

  [ m , where m = 4 k ]	[ n = m 2 /4 - 1 ]	[ n +2]
  4	3	5
  8	15	17
  12	35	37
  16	63	65
  20	99	101
  24	143	145
  28	195	197
  32	255	257	... etc ...

• other triples:
  20, 21, 29 -- n = 21: n ² + 16 n + 64 = ( n +8) ²
  28, 45, 53 -- n = 45: n ² + 16 n + 64 = ( n +8) ²
  36, 77, 85 -- n = 77: n ² + 16 n + 64 = ( n +8) ²
  33, 56, 65 -- n = 56: n ² + 18 n + 81 = ( n +9) ²
  39, 80, 89 -- n = 80: n ² + 18 n + 81 = ( n +9) ²
  48, 55, 73 -- n = 55: n ² + 36 n + 324 = ( n +18) ²
  60, 91, 109 -- n = 91: n ² + 36 n + 324 = ( n +18) ²
  65, 72, 97 -- n = 72: n ² + 50 n + 625 = ( n +25) ²
  etc ...
  
  
  
  
  
  
  
  
  

• Conway and Guy. The Book of Numbers , chapter 6