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Product of an integer with itself

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 3^(2) 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 n^(2), 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 n^(2). 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). In the real number system, square numbers are non negative. A non negative integer is a square number when its square root is again an integer. For example, so 9 is a square number. 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 n^(2), with 1=0^(2) = 0 being the zeroth 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, for example,
Starting with 1, there are square numbers up to and including m, where the expression represents the floor of the number x.

Squares of even numbers are even, and are divisible by 4, since 1=(2n)^(2) = 4n^(2). Squares of odd numbers are odd, and are congruent to 1 modulo 8, since 1=(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 2^(n) differ by an amount containing an odd factor, the only perfect square of the form 2^(n) − 1 is 1, and the only perfect square of the form 2^(n) + 1 is 9.

If the number is of the form m5 where m represents the preceding digits, its square is n25 where 1=n = m(m + 1) and represents digits before 25. For example, the square of 65 can be calculated by 1=n = 6 × (6 + 1) = 42 which makes the square equal to 4225. If the number is of the form m0 where m represents the preceding digits, its square is n00 where 1=n = m^(2). For example, the square of 70 is 4900. If the number has two digits and is of the form 5m where m represents the units digit, its square is aabb where 1=aa = 25 + m and 1=bb = m^(2). For example, to calculate the square of 57, 1=m = 7 and 1=25 + 7 = 32 and 1=7^(2) = 49, so 1=57^(2) = 3249. If the number ends in 5, its square will end in 5; similarly for ending in 25, 625, 0625, 90625, 8212890625, etc. If the number ends in 6, its square will end in 6, similarly for ending in 76, 376, 9376, 09376, 1787109376. For example, the square of 55376 is 3066501376, both ending in 376. (The numbers 5, 6, 25, 76, etc. are called automorphic numbers. They are sequence A003226 in the OEIS.) In base 10, the last two digits of square numbers follow a repeating pattern mirrored symmetrical around multiples of 25. In the example of 24 and 26, both 1 off from 25, 1=24^(2) = 576 and 1=26^(2) = 676, both ending in 76. In general, An analogous pattern applies for the last 3 digits around multiples of 250, and so on. As a consequence, of the 100 possible last 2 digits, only 22 of them occur among square numbers (since 00 and 25 are repeated).