Happy birthday to the United States! It’s one of the younger countries on the planet and yet has still managed to have one of the oldest continuous systems of government. Not too shabby. Here’s hoping our current wobbles get straightened out and our next few Independence Days occur under more pleasant conditions.

I’m not sure what function might fit very well in the context of this holiday, but I’ll make sort of a stretch and do a function about the Pythagorean theorem. The Pythagorean theorem is a very ancient discovery with numerous different methods of proof. One of them was discovered by James Garfield, who would later become a president of the United States. I don’t think many of our 44 presidents have made independent nontrivial mathematical discoveries, so good for him. Here’s a figure from his proof:

The Pythagorean theorem states that for the sides of a right triangle A, B, and C, the identity A^2 + B^2 = C^2 will be satisfied. You may be familiar with the Scarecrow’s statement of the theorem in the Wizard of Oz, though he actually butchers it somewhat and what he says isn’t actually true.

For certain side lengths, you’ll find that the lengths can be specified with pure integers. For example, the triple {3, 4, 5} is a so-called Pythagorean triple because 3^2 + 4+2 = 5^2. Go ahead, try it out. There’s an infinite number of these triples, and they can be generated with our Sunday Function:

Given any two integers m and n (with m > n), and this formula will generate three numbers a, b, and c which are Pythagorean triples. For instance, with n = 1776 and m = 2010, you’ll get the triple {885924, 7139520, 7194276}.

Ok, enough math for the day. Now go cook some burgers, celebrate your freedoms, and try not to lose any fingers to fireworks!