Yesterday's post ended with the statement of the generating function for Pythagorean triples. Take any two distinct whole numbers j and k and let j be the larger. Here are the formulas for a, b and c.
a = j² - k²
b = 2jk
c = j² + k²
Example #1: j = 2, k = 1
a = 4-1 = 3
b = 2 × 2 × 1 = 4
c = 4+1 = 5
Example #2: j = 3, k = 1
a = 9-1 = 8
b = 2 × 3 × 1 = 6
c = 9+1 = 10
Example #3: j = 3, k =2
a = 9-4 = 5
b = 2 × 3 × 2 =12
c = 9+4 = 13
Not surprisingly, using the three smallest pairs around (2, 1) (3, 1) and (3, 2), we get the three smallest Pythagorean triples. But notice that (3, 1) generates 8-6-10 which is not a relatively prime triple, since all the numbers are even. Looking at this carefully, we see that if both j and k are odd, this will make a, b and c even.
a = odd² - odd² = even
b = 2×odd×odd = even
c = odd² + odd² = even
To generate a relatively prime Pythagorean triple, we need two things to be true about j and k.
1. j and k must be relatively prime.
2. j - k must be odd. (Another way to state this is one is odd and the other even.)
Tomorrow, a few more rules about the relatively prime Pythagorean triples.
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