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Square number
From Wikipedia In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. (In other words, a number whose square root is an integer.) So for example, 9 is a square number since it can be written as 3 × 3. If rational numbers are included, then the ratio of two square integers is also a square number (e.g. 2/3 × 2/3 = 4/9).
The number m is a square number if and only if one can arrange m points in a square.
The first 50 squares A000290 are:
- 12 = 1
- 22 = 4
- 32 = 9
- 42 = 16
- 52 = 25
- 62 = 36
- 72 = 49
- 82 = 64
- 92 = 81
- 102 = 100
- 112 = 121
- 122 = 144
- 132 = 169
- 142 = 196
- 152 = 225
- 162 = 256
- 172 = 289
- 182 = 324
- 192 = 361
- 202 = 400
- 212 = 441
- 222 = 484
- 232 = 529
- 242 = 576
- 252 = 625
- 262 = 676
- 272 = 729
- 282 = 784
- 292 = 841
- 302 = 900
- 312 = 961
- 322 = 1024
- 332 = 1089
- 342 = 1156
- 352 = 1225
- 362 = 1296
- 372 = 1369
- 382 = 1444
- 392 = 1521
- 402 = 1600
- 412 = 1681
- 422 = 1764
- 432 = 1849
- 442 = 1936
- 452 = 2025
- 462 = 2116
- 472 = 2209
- 482 = 2304
- 492 = 2401
- 502 = 2500
n2 equals to the sum of the first n odd numbers ([math]\displaystyle{ n^2 = 2(n-1)^2-(n-2)^2+2 }[/math]). A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.
An easy way to find square numbers is to find two numbers which have a mean of it, 212:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22x20=440+12=441. This works because of the identity
- (x-y)(x+y)=x2–y2
known as the difference of two squares. Thus (21–1)(21+1)=212–12=440, if you work backwards.
