- Square number
In

mathematics , a**square number**, sometimes also called a, is anperfect square integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers arenon-negative . Another way of saying that a (non-negative) number is a square number, is that itssquare root is again an integer. For example, √9 = 3, so 9 is a square number.A positive integer that has no perfect square

divisor s except 1 is called square-free.The usual notation for the formula for the square of a number "n" is not the product "n" × "n", but the equivalent

exponentiation "n"^{2}, usually pronounced as "n" squared". For a non-negative integer "n", the "n"th square number is "n"^{2}, with 0^{2}= 0 being thezeroth square. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square (e.g., 4/9 = (2/3)^{2}).Starting with 1, there are ⌊√"m"⌋ square numbers up to and including "m".

**Examples**The first 50 squares of

natural number s OEIS|id=A000290 are::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:

The formula for the "n"th square number is "n"

^{2}. This is also equal to the sum of the first "n"odd number s:$n^2\; =\; sum\_\{k=1\}^n(2k-1)$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, 5^{2}= 25 = 1 + 3 + 5 + 7 + 9.The "n"th square number can be calculated from the previous two by doubling the ("n" − 1)-th square, subtracting the ("n" − 2)-th square number, and adding 2, because "n"

^{2}= 2("n" − 1)^{2}− ("n" − 2)^{2}+ 2. For example, 2×5^{2}− 4^{2}+ 2 = 2×25 − 16 + 2 = 50 − 16 + 2 = 36 = 6^{2}.A square number is also the sum of two consecutive

triangular number s. The sum of two consecutive square numbers is acentered square number . Every odd square is also acentered octagonal number .Lagrange's four-square theorem states that any positive integer can be written as the sum of 4 or fewer perfect squares. Three squares are not sufficient for numbers of the form 4^{"k"}(8"m" + 7). A positive integer can be represented as a sum of two squares precisely if itsprime factorization contains no odd powers of primes of the form 4"k" + 3. This is generalized byWaring's problem .A square number can only end with digits 00,1,4,6,9, or 25 in base 10, as follows:

#If the last digit of a

number is 0, its square ends in 00 and the precedingdigit s must also form a square.

#If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.

#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 number formed by its preceding digits must be divisible by four.

#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 of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.A square number cannot be a

perfect number .**Easy ways to calculate square numbers**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: 22×20 = 440 + 1^{2}= 441. This works because of the identity:("x" − "y")("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.**pecial cases*** If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m*(m+1) and represents digits before 25. For example the square of 65 can be calculated by n=6*(6+1)=42 which makes the square equal to 4225.

* If the number is of the form m0 where m represents the preceding digits, its square is n00 where n = m^{2}. For example the square of 70 is 4900.**Odd and even square numbers**Squares of even numbers are even, since (2"n")

^{2}= 4"n"^{2}.Squares of odd numbers are odd, since (2"n" + 1)

^{2}= 4("n"^{2}+ "n") + 1.It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.

**Chen's theorem**Chen Jingrun showed in 1975 that there always exists a number "P" which is either a prime or product of two primes between "n"^{2}and ("n" + 1)^{2}. See alsoLegendre's conjecture .**ee also***

Integer square root

*Methods of computing square roots

*Quadratic residue

*Polygonal number

*triangular square number

*Euler's four-square identity

*Automorphic number **References***MathWorld|urlname=SquareNumber|title=Square Number

**Further reading***Conway, J. H. and Guy, R. K. "The Book of Numbers". New York: Springer-Verlag, pp. 30-32, 1996. ISBN 0-387-97993-X

**External links*** [

*http://www.learntables.co.uk/square_numbers/ Learn Square Numbers*] . Practice square numbers up to 144 with this children's multiplication game

* Dario Alpern, [*http://www.alpertron.com.ar/FSQUARES.HTM Sum of squares*] . A Java applet to decompose a natural number into a sum of up to four squares.

* [*http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1296&bodyId=1433 Fibonacci and Square Numbers*] at [*http://mathdl.maa.org/convergence/1/ Convergence*]

* [*http://www.naturalnumbers.org/psquares.html The first 1,000,000 perfect squares*] Includes a program for generating perfect squares up to 10^15.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Square number**— Square Square (skw[^a]r), a. 1. (Geom.) Having four equal sides and four right angles; as, a square figure. [1913 Webster] 2. Forming a right angle; as, a square corner. [1913 Webster] 3. Having a shape broad for the height, with rectilineal and… … The Collaborative International Dictionary of English**square number**— ► NOUN ▪ the product of a number multiplied by itself, e.g. 1, 4, 9, 16 … English terms dictionary**square number**— a number that is the square of another integer, as 1 of 1, 4 of 2, 9 of 3, etc. [1550 60] * * * square number noun A number the square root of which is an integer • • • Main Entry: ↑square * * * square number, the product of a number multiplied… … Useful english dictionary**square number**— /skwɛə ˈnʌmbə/ (say skwair numbuh) noun a number which is the square of some integer number, as 1, 4, 9, 16, 25, etc., with respect to 1, 2, 3, 4, 5, etc … Australian English dictionary**square number**— a number that is the square of another integer, as 1 of 1, 4 of 2, 9 of 3, etc. [1550 60] * * * … Universalium**square number**— noun the product of a number multiplied by itself, e.g. 1, 4, 9, 16 … English new terms dictionary**Centered square number**— In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each… … Wikipedia**Square root of a number**— Square Square (skw[^a]r), a. 1. (Geom.) Having four equal sides and four right angles; as, a square figure. [1913 Webster] 2. Forming a right angle; as, a square corner. [1913 Webster] 3. Having a shape broad for the height, with rectilineal and… … The Collaborative International Dictionary of English**Square**— (skw[^a]r), a. 1. (Geom.) Having four equal sides and four right angles; as, a square figure. [1913 Webster] 2. Forming a right angle; as, a square corner. [1913 Webster] 3. Having a shape broad for the height, with rectilineal and angular rather … The Collaborative International Dictionary of English**Square foot**— Square Square (skw[^a]r), a. 1. (Geom.) Having four equal sides and four right angles; as, a square figure. [1913 Webster] 2. Forming a right angle; as, a square corner. [1913 Webster] 3. Having a shape broad for the height, with rectilineal and… … The Collaborative International Dictionary of English