I find completely useless number theory fun.  Amicable numbers!  Who-hoo—and utterly useless!

In the spring of 2005, while I was spending a week with my grandmother, I was playing with various functions and a knitting project.  I quickly discovered that the difference between consecutive squares is consecutive odd numbers.  (For example, the first five squares of my pattern were 1, 4,  9, 16, and 25. The difference between 4 and 1 is 3, between 9 and 4 is 5, between 16 and 9 is 7, and between 25 and 16 is 9—3, 5, 7, 9.)

When I came home, I told my then-boyfriend of my discovery and he said, "It breaks down at bigger numbers."  I argued with him until he both started proving it.  I proved it by drawing squares on paper, and he proved it algebraically.

So now I'm interested in figuring out more patterns about exponents.  I'm sure I could do some research and find out that someone's already done the work but...where's the fun in that?