Happy Pythagorean Triple Day (5-12-13)
We all know (or should know if you graduated Jr high) what the Pythagorean Theorem is: for a right triangle abc, a^2+b^2=c^2. For many this is one of the first natural laws that we learn. For many more they don’t even learn this and wind up working at Walmart for the rest of their adult lives while their teeth rot out of their heads from all the diet coke they drink. But that’s a different story…
A Pythagorean Triple is a special case of the Pythagorean Theorem, where all three sides are of integer length. There are infinitely many Pythagorean Triples (that means that there’s more of them then there are mouth breathers at a Nickleback concert) since any integer scaling of a Pythagorean Triple produces yet another Pythagorean Triple: (ka)^2+(kb)^2=(kc)^2 where k is an integer.
Combine that with the fact that there are infinitely many Primitive Pythagorean Triples (a,b and c are all co-prime) results in a huge number of triples.
A cool way to generate a Pythagorean Triple is to use Euclid’s Formula: a=m^2-n^2; b=2mn; c=m^2+n^2, where m>n. This generates a Primitive Pythagorean Triple if m & n are co-prime, and one of them is odd and the other is even.
So today is May 12, 2013 (5-12-13) and 5^2+12^2=13^2; 25+144=169. The next Pythagorean Triple is December 5, 2013 (12-5-13 works just as well as 5-12-13). After that we have to wait until August 15, 2017 (8-15-17).