Square number

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".

Hence, a square with side length n has area n2.

A positive integer that has no square divisors except 1 is called square-free.

For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zeroth one.

The concept of square can be extended to some other number systems.

square numbers up to and including m, where the expression

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).

There are several recursive methods for computing square numbers.

For example, The square minus one of a number m is always the product of

Since a prime number has factors of only 1 and itself, and since m = 2 is the only non-zero value of m to give a factor of 1 on the right side of the equation above, it follows that 3 is the only prime number one less than a square (3 = 22 − 1).

This is the difference-of-squares formula, which can be useful for mental arithmetic: for example, 47 × 53 can be easily computed as 502 − 32 = 2500 − 9 = 2491.

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 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.

In base 10, a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows: In base 12, a square number can end only with square digits (like in base 12, a prime number can end only with prime digits or 1), that is, 0, 1, 4 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).

[citation needed] All such rules can be proved by checking a fixed number of cases and using modular arithmetic.

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).

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.)

The first values of these sums, the square pyramidal numbers, are: (sequence A000330 in the OEIS) 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.

, 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.

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.