Three positive integers that describe a right-angled triangle:
i ^{ 2 } + j ^{ 2 } = k ^{ 2 }
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.
[ 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 ... |
[ 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 ... |