Let’s listen in on a conversation between siblings divided by math, so to speak:

“That’s amazing!” said Mathophila. “The squares of three consecutive positive whole numbers add up to the same number as the squares of the next two numbers.”

“Oh, no big deal, that must happen all the time,” replied her brother, Innumeratus.

“I don’t know, it seems like an amazing coincidence to me!” retorted Mathophila.

Find the smallest collection of numbers with the described property, and settle the argument between the siblings: determine whether there are infinitely many sets of numbers as described by Mathophila, or a finite number (and if so, figure out exactly how many solutions there are).