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Square number
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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 n2, usually pronounced as "n squared". For a non-negative integer n, the nth square number is n2, 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.
ExamplesThe first 50 squares of natural numbers are:
PropertiesThe number m is a square number if and only if one can arrange m points in a square:
| 12 = 1 | | | 22 = 4 | | | 32 = 9 | | | 42 = 16 | | | 52 = 25 | |
The formula for the nth square number is n2. 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, 52 = 25 = 1 + 3 + 5 + 7 + 9.
The nth 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 n2 = 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 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 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.
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) = x2 − y2
known as the difference of two squares. Thus (21–1)(21 + 1) = 212 − 12 = 440, if you work backwards.
A square number cannot be a perfect number.
Odd and even square numbersSquares of even numbers are even, since (2n)2 = 4n2.
Squares of odd numbers are odd, since (2n + 1)2 = 4(n2 + n) + 1.
It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.
Chen's theoremChen Jingrun showed in 1975 that there always exists a number P which is either a prime or product of two primes between n2 and (n + 1)2. See also Legendre's conjecture.
See also
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- . Practice square numbers up to 144 with this childrens multiplication game
- Dario Alpern, . A Java applet to decompose a natural number into a sum of up to four squares.
- at
- Includes a program for generating perfect squares up to 10^15.
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