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 equals 32 and can be written as 3 × 3.
The usual notation 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. The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with side length n has area n2. If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers).
The difference between any perfect square and its predecessor is given by the identity n2 − (n − 1)2 = 2n − 1. Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is, n2 = (n − 1)2 + (n − 1) + n.
The number m is a square number if and only if one can arrange m points in a square:
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:
There are several recursive methods for computing square numbers. For example, the nth square number can be computed from the previous square by n2 = (n − 1)2 + (n − 1) + n = (n − 1)2 + (2n − 1). 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 (except 0) 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.
In base 10, a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows:
Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example). All such rules can be proved by checking a fixed number of cases and using modular arithmetic.
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.
0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201...
The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains Galileo's law of odd numbers: if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length. From s = ut + 1/2at2, for u = 0 and constant a (acceleration due to gravity without air resistance); so s is proportional to t2, and the distance from the starting point are consecutive squares for integer values of time elapsed.
All fourth powers, sixth powers, eighth powers and so on are perfect squares.
Squares of even numbers are even, and are divisible by 4, since (2n)2 = 4n2. Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1)2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8.
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.