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, 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 are non-negative. Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.

A positive integer that has no perfect square divisors 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 0² = 0 being the zeroth 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 numbers OEIS|id=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

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 numbers:$n^2\; =\; sum\_\{k=1\}^n(2k-1)$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, 5² = 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, 2×5² − 4² + 2 = 2×25 − 16 + 2 = 50 − 16 + 2 = 36 = 6².

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. 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 its prime factorization contains no odd powers of primes of the form 4"k" + 3. This is generalized by Waring'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 preceding digits 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, 21²:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22×20 = 440 + 1² = 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) = 21² − 1² = 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 = m². 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 also Legendre'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.