pell's equation

I now learn that I studied pell's diophantine equation in 7th-grade or so, though i didn't know it was called that. I was messing with newton's method of calculating square roots, and doing everything with exact fractions, and noticed that some intermediate fraction (the correction term) in each newton iteration tended be be something like 1/n. can be seen by seeding sqrt(2) with an initial guess of 3/2, or sqrt(3) with an inital guess of 2.