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 squarefree.
Contents
Examples
The first 51 squares (sequence A000290 in OEIS) are: 0^{2} = 0
 1^{2} = 1
 2^{2} = 4
 3^{2} = 9
 4^{2} = 16
 5^{2} = 25
 6^{2} = 36
 7^{2} = 49
 8^{2} = 64
 9^{2} = 81
 10^{2} = 100
 11^{2} = 121
 12^{2} = 144
 13^{2} = 169
 14^{2} = 196
 15^{2} = 225
 16^{2} = 256
 17^{2} = 289
 18^{2} = 324
 19^{2} = 361
 20^{2} = 400
 21^{2} = 441
 22^{2} = 484
 23^{2} = 529
 24^{2} = 576
 25^{2} = 625
 26^{2} = 676
 27^{2} = 729
 28^{2} = 784
 29^{2} = 841
 30^{2} = 900
 31^{2} = 961
 32^{2} = 1024
 33^{2} = 1089
 34^{2} = 1156
 35^{2} = 1225
 36^{2} = 1296
 37^{2} = 1369
 38^{2} = 1444
 39^{2} = 1521
 40^{2} = 1600
 41^{2} = 1681
 42^{2} = 1764
 43^{2} = 1849
 44^{2} = 1936
 45^{2} = 2025
 46^{2} = 2116
 47^{2} = 2209
 48^{2} = 2304
 49^{2} = 2401
 50^{2} = 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 n^{2}. This is also equal to the sum of the first n odd numbers (), 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, 5^{2} = 25 = 1 + 3 + 5 + 7 + 9.
The nth square number can be calculated from the previous two by adding the n1th square to itself, subtracting the n2th square number, and adding 2 (). For example, 2×5^{2}  4^{2} + 2 = 2×25  16 + 2 = 50  16 + 2 = 36 = 6^{2}.
It is often also useful to note that the square of any number can be represented as the sum 1 + 1 + 2 + 2 + ... + n1 + n1 + n. For instance, the square of 4 or 4^{2} 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 = 50^{2} + 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 foursquare 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 4^{k}(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:
 If the last digit of a number is 1 or 9, its square ends in 1 and the preceding digit must be even.
 If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
 If the last digit of a number is 3 or 7, its square ends in 9 and the preceding digit must also be even.
 If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
 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, 21^{2}:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22x20=440+1^{2}=441. This works because of the identity
 (xy)(x+y)=x^{2}–y^{2}
known as the difference of two squares. Thus (21–1)(21+1)=21^{2}–1^{2}=440, if you work backwards.
A square number can't be a perfect number.
External links
 http://www.alpertron.com.ar/FSQUARES.HTM is a Java applet that decomposes a natural number into a sum of up to four squares.