Squares

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.