# 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 ...