In an attempt to get back into shape, I’ve started running. While I run, my mind has a tendency to wander, and when the going gets tough, I start doing math in my head to take my mind off the raging inferno in my lungs.
The other day, I came up with a formula. I was squaring integers in sequence, and I figured out that the distance between squares of consecutive integers is derivable. Put another way, if you know the square of one integer, there’s a pattern — a simple trick — to finding the square of the next integer.
If you know this:
14 x 14 = 196
Simply double 14, add 1, and add to the previous result to get the square of the next integer — 15.
(14 x 14) + (14 x 2) + 1 = 225 = 15 x 15
This can be represented by:
x2 = (x-1)2 + 2(x - 1) + 1
x2 + 2x + 1 = (x + 1)2
It works on small numbers:
(1 x 1) + (1 x 2) + 1 = 4 = 2 x 2
And bigger numbers:
(165 x 165) + (165 x 2) + 1 = 27,556 = 166 x 166
Now, I’m not naive enough to think that I’ve stumbled on Fermat’s Theorem or anything. I’m fully aware this is simple math that has no-doubt been documented thousands of times.
But does it have a name? Is there some kind of principle here that I inadvertantly stumbled on? I’m just curious. If anyone has a mathmetician friend, send him this link.