Pythagorean Triples

Let’s find special right triangles where all three right triangles had whole numbers for sides, namely:

      9 : 40 : 41,

     12 : 35 : 37,

       7 : 24 : 25 and

      5 : 12 : 13 triangle (actually 10:24:26, but that's similar to a 5:12:13 triangle). And no doubt you already know about the 3:4:5 right triangle.

So, are there other special right triangles whose sides are all whole numbers? Yes, and they've been studied for a long time.

Three numbers a, b, and c such that a² + b² = c² are said to form a Pythagorean triple, in honor of Pythagoras. He lived about 550 B.C. and probably know quite a few of them.

But the Old Babylonians of about 1800 B.C. know them all, and many were known in other ancient civilizations such as China and India.

Here's a modern paraphrase of Euclid. Take any two odd numbers m and n, with m <  n, and relatively prime (that is, no common factors).

Let a = m• n, let

b = (n² – m²)/2, and let c = (n² + m²)/2. Then a : b : c is a Pythagorean triple. For instance, if you take m = 1, and n = 3, then you get the smallest Pythagorean triple

3 : 4 : 5.