In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an 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 nonnegative. Another way of saying that a (nonnegative) 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 squarefree.
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 nonnegative integer n, the nth square number is n^{2}, with 0^{2} = 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 square numbers up to and including m.
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The squares (sequence A000290 in OEIS) below 50^{2} are:
The difference between any perfect square and its predecessor is given by the following identity,
The number m is a square number if and only if one can arrange m points in a square:
m = 1^{2} = 1  
m = 2^{2} = 4  
m = 3^{2} = 9  
m = 4^{2} = 16  
m = 5^{2} = 25 
The expression for the nth square number is n^{2}. 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 cyan). The formula follows:
So for example, 5^{2} = 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 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 numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.
Lagrange's foursquare 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}(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:
A square number cannot be a perfect number.
Squares of even numbers are even, since (2n)^{2} = 4n^{2}.
Squares of odd numbers are odd, since (2n + 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.
A square number, sometimes also called a perfect square, is the result of an integer multiplied by itself. 1, 4, 9, 16, 25, ... are the first few square numbers.
The square root of a perfect square is always an integer.
There are infinitely more examples that are possible. These are just the squares of natural numbers up to 10.
