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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.

00502500
11492401512601
24482304522704
39472209532809
416462116542916
525452025553025
636441936
749431849
864421764
981411681
10100401600
11121391521
12144381444
13169371369
14196361296
15225351225
16256341156
17289331089
18324321024
1936131961
2040030900
2144129841
2248428784
2352927729
2457626676
25625

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.

 

  See Also

  Algorithm, The
  History and Other Stuff
  Intermediates
  Logarithmic Forms
  Multiplication Table, The
  Powers of N
  Square Roots
  Sums of Squares
  Triangular Numbers

@ July (2005)