Square number


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.


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