Babylonian Pythagorean Triplets
Question: the Babylonians catalogued many Pythagorean triplets of numbers (centuries before Pythagoras!) including the enormous 3,367 : 3,456 : 4,825. Obviously, they didn't check every triplet of integers, even plausible looking ones, up to that value. How could they possibly have come up with that set?
Suppose they did discover a few sets of integers by trial and error, say 3 : 4: 5, 5 : 12 : 13, 7 : 24 : 25, 8 : 15 : 17. We'll assume they didn't count 6 : 8 : 10, and other triplets where all three numbers have a common factor, since that's not really anything new.
Now they contemplate their collection of triplets. Remember, they're focused on sums of squares here. So, they probably noticed that all their triplets had a remarkable common property: the largest member of each triplet (whose square is of course the sum of the squares of the other two members) is in fact itself a sum of two squares! Check it out: 5 = 22 + 12, 13 = 32 + 22, 25 = 42 + 32, 17 = 42 + 12.
Staring at the triplets a little longer they might have seen that once you express the largest member as a sum of two squares, one of the other two members of the triplet is the difference of the same two squares! That is, 3 = 22 - 12, 5 = 32 -22, 7 = 42 - 32, 15 = 42 - 12.
How does the third member of the triplet relate to the numbers we squared and added to get the largest member? It's just twice their product! That is, 4 = 2x2x1, 12 = 2x3x2, 24 = 2x4x3, 8 = 2x4x1.
This at least suggests a way to manufacture larger triplets, which can then be checked by multiplication. We need to take the squares of two numbers that don't have a common factor (otherwise, all members of the triplet will have that factor). an even power of 2, the second an even power of 3. (The order in the difference term could of course be reversed, depending on which one's bigger).
In fact, notice that 5 : 12 : 13 and 7 : 24 : 25 are already of this form, with 22, 32 and 24, 32. What about 26, 32? That gives 48 : 55 : 73. Try another: 26, 34 gives 17 : 144 : 145. But why stop there? Let's try something bigger: 212, 36. That gives the triplet 3,367 : 3,456 : 4,825. Not so mysterious after all.