Home

> urticator.net
Search

> Domains
Glue
Stories

Computers
Driving
Games
Humor
Law
Math
> Numbers
Science

Powers and Fractions
Decimal Expansions
Repeat Length Again
Number Maze
Primes
Divisibility
Intrinsic Nature of Primes
Other Topics
Other Topics (2)

 > Squares
Triangular Numbers
Intermediates
Square Roots
Sums of Squares
Differences

## Squares

I'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.

 0 0 50 2500 1 1 49 2401 51 2601 2 4 48 2304 52 2704 3 9 47 2209 53 2809 4 16 46 2116 54 2916 5 25 45 2025 55 3025 6 36 44 1936 7 49 43 1849 8 64 42 1764 9 81 41 1681 10 100 40 1600 11 121 39 1521 12 144 38 1444 13 169 37 1369 14 196 36 1296 15 225 35 1225 16 256 34 1156 17 289 33 1089 18 324 32 1024 19 361 31 961 20 400 30 900 21 441 29 841 22 484 28 784 23 529 27 729 24 576 26 676 25 625

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.