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Patterns in a2 + b2 = c2 (March 2002)

There are an infinite number of integral solutions to the Pythagorean formula a2 + b2 = c2. Here's a plot of the integral solutions for a < 300 and b < 200.

Point plot of a vs. b where a*a + b*b = c*c for integers a, b, c

As you can see, there are clear patterns in the distribution. This month, I'm suggesting that you look for the patterns. Explore freely. To help you out, here's another plot. This time with some lines to guide your eyes.

Point plot of a vs. b with suggestive curves addedD

Are there fundamental relationships underlying these lines? If so, what are they?

Send all responses to my email address is mathrec at this domain.

Thanks,
Steve