About This Site
Powers and Fractions
> Notes About Squares
Repeat Length Again
Intrinsic Nature of Primes
Other Topics (2)
Sums of Squares
SquaresI'm sure you know this, but let me say it anyway. If you have a number n (1 2 3 4 … ), you can square it to get another number n2 (“n squared”, 1 4 9 16 … ). The concept is so familiar, you can almost forget where the name “square” came from.
I know a lot of squares reflexively, by association rather than calculation, but the upper limit varies. When I wrote Powers of N, the limit was 27; but right now it's 32, since I happen to know 282 = 784, and the rest are easy. With a quick review I can push the limit up to 41, and with only a bit of effort I can push it to 50 or 55. That's a fairly recent development, though, helped along by learning the products up to 2100. Only the prime squares count as true products, but the other squares are useful as reference points; for example, 41×43 = 422-1.
Anyway, here's a table of all the squares I'm interested in.
Now, here's a funny thing. As long as I can remember, I've known the squares 242 = 576 and 262 = 676; I found them memorable because by strange coincidence they had the same last digits. Yes, believe it or not, it wasn't until a year or two ago that I noticed the pattern.
(25+d)2 = (25-d)2 + 100d
As d approaches 25, 25+d starts to look more like 50-n.
(50-n)2 = 2500 - 100n + n2
That's “another” pattern I'd never noticed, even though I've known about the striking number 492 = 74 = 2401 essentially forever.
Finally, as you can see in the table, the pattern continues for numbers over 50.
(50+n)2 = 2500 + 100n + n2
The whole thing is really just a consequence of two little facts: (-x)2 = x2, and (x+2m)2 ≡ x2 mod 4m.
History and Other Stuff
Multiplication Table, The
Powers of N
Sums of Squares
@ July (2005)