“Pythagorean triples” are integer solutions to the Pythagorean Theorem, a2 + b2 = c2. I like “triplets,” but “triples” seems to be the favored term. For a right triangle, the c side is the hypotenuse, the side opposite the right angle. The a side is the shorter of the two sides adjacent to the right angle. The first rules that I became aware of for determining a subset of Pythagorean triplets are as follows:
- Every odd number is the a side of a Pythagorean triplet.
- The b side of a Pythagorean triplet is simply (a2 – 1) / 2.
- The c side is b + 1.
Here, a and c are always odd; b is always even. These relationships hold because the difference between successive square numbers is successive odd numbers. Every odd number that is itself a square (and the square of every odd number is an odd number) thus makes for a Pythagorean triplet. Thus, the square of 7, 49, is the difference between576, the square of 24, and 625, the square of 25, giving us the triplet 7,24,25. Similarly, the square of 23, 529, is the difference between 69696, the square of 264, and 70225, the square of 265, giving us the triplet 23,264,265.
The simplest triplet in the table, 1,0,1, is not a triangle, and a>b, but it is a solution to the Pythagorean Theorem — a very trival one since n,0,n, where n is any number, works just as well (using 0 would even allow us to break Fermat’s Last Theorem).