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In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3 × 3.
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". The name square number comes from the name of the shape; see below.
For a non-negative integer n, the nth square number is n2, with 02 = 0 being the 0-th one. 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 square numbers up to and including m, where the expression represents the floor of the number x.
The difference between any perfect square and its predecessor is given by the identity . Equivalently, it is possible to count up square numbers by adding together the last square, the last square's root, and the current root, that is, .
The number m is a square number if and only if one can compose a square of m equal (lesser) squares:
|m = 12 = 1|
|m = 22 = 4|
|m = 32 = 9|
|m = 42 = 16|
|m = 52 = 25|
|Note: White gaps between squares serve only to improve visual perception.|
There must be no gaps between actual squares.
The expression 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 (shown in magenta). The formula follows:
So for example, 52 = 25 = 1 + 3 + 5 + 7 + 9.
There are several recursive methods for computing square numbers. For example, the nth square number can be computed from the previous square by . Alternatively, 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,
Another property of a square number is that it has an odd number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.
Lagrange's four-square theorem states that any positive integer can be written as the sum of four 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 end only with digits 0, 1, 4, 6, 9, or 25 in base 10, as follows:
In base 16, a square number can end only with 0, 1, 4 or 9 and
In general, if a prime p divides a square number m then the square of p must also divide m; if p fails to divide m∕p, then m is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number m is a square number if and only if, in its canonical representation, all exponents are even.
Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k, if k2 − m is the square of an integer n then k − n divides m. (This is an application of the factorization of a difference of two squares.) For example, 1002 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from k − n to k + n where k covers some range of natural numbers k ≥ √.
A square number cannot be a perfect number.
The sum of the series of power numbers
can also be represented by the formula
The first terms of this series (the square pyramidal numbers) are:
All fourth powers, sixth powers, eighth powers and so on are perfect squares.
Squares of even numbers are even (and in fact divisible by 4), 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.
As all even square numbers are divisible by 4, the even numbers of the form 4n + 2 are not square numbers.
As all odd square numbers are of the form 4n + 1, the odd numbers of the form 4n + 3 are not square numbers.
Squares of odd numbers are of the form 8n + 1, since (2n + 1)2 = 4n(n + 1) + 1 and n(n + 1) is an even number.
Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8. The difference between 1 and any higher odd perfect square always is eight times a triangular number, while the difference between 9 and any higher odd perfect square is eight times a triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of 2n differ by an amount containing an odd factor, the only perfect square of the form 2n - 1 is 1, and the only perfect square of the form 2n + 1 is 9.