# Square number

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 (e.g. 2/3 × 2/3 = 4/9).

A positive integer that has no perfect square divisors except 1 is called square-free.

## Examples

The first 51 squares (sequence A000290 in OEIS) are: 02 = 0

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

## Properties

The number m is a square number if and only if one can arrange m points in a square:

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The formula for the nth square number is n2. This is also equal to the sum of the first n odd numbers (${\displaystyle n^{2}=\sum _{1}^{n}(2n-1)}$), as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (marked as '+'). So for example, 52 = 25 = 1 + 3 + 5 + 7 + 9.

The nth square number can be calculated from the previous two by adding the n-1th square to itself, subtracting the n-2th square number, and adding 2 (${\displaystyle n^{2}=2(n-1)^{2}-(n-2)^{2}+2}$). For example, 2×52 - 42 + 2 = 2×25 - 16 + 2 = 50 - 16 + 2 = 36 = 62.

It is often also useful to note that the square of any number can be represented as the sum 1 + 1 + 2 + 2 + ... + n-1 + n-1 + n. For instance, the square of 4 or 42 is equal to 1 + 1 + 2 + 2 + 3 + 3 + 4 = 16. This is the result of adding a column and row of thickness 1 to the square graph of three (like a tic tac toe board). You add three to the side and four to the top to get four squared. This can also be useful for finding the square of a big number quickly. For instance, the square of 52 = 502 + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.

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.

Lagrange's four-square theorem states that any positive integer can be written as the sum of 4 or fewer perfect squares. 3 squares are not sufficient for numbers of the form 4k(8m + 7). A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4k+3. This is generalized by Waring's problem.

A square number can only end with digit 0,1,4,5,6,9, following these rules:

1. If the last digit of a number is 1 or 9, its square ends in 1 and the preceding digit must be even.
2. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
3. If the last digit of a number is 3 or 7, its square ends in 9 and the preceding digit must also be even.
4. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
5. If the last digit is 5 or 0, its square ends in the last digit of the root and the preceding digits must be square.

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.

A square number can't be a perfect number.