In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and This article refers to the REM live recording For the mathematical term see Perfect square. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Algebra, the square of a number is that number multiplied by itself So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. A negative number is a Number that is less than zero, such as −2 Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose For example, √9 = 3, so 9 is a square number.

A positive integer that has no perfect square divisors except 1 is called square-free. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Mathematics, an element r of a Unique factorization domain R is called square-free if it is not divisible by a non-trivial square

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". For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zeroth square. The zeroth item is the initial item of a zero -based sequence (that is a sequence which is numbered beginning from zero rather than one such as the non-negative integers (see 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 ⌊√m⌋ square numbers up to and including m.

## Examples

The first 50 squares of natural numbers (sequence A000290 in OEIS) are:

12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400
212 = 441
222 = 484
232 = 529
242 = 576
252 = 625
262 = 676
272 = 729
282 = 784
292 = 841
302 = 900
312 = 961
322 = 1024
332 = 1089
342 = 1156
352 = 1225
362 = 1296
372 = 1369
382 = 1444
392 = 1521
402 = 1600
412 = 1681
422 = 1764
432 = 1849
442 = 1936
452 = 2025
462 = 2116
472 = 2209
482 = 2304
492 = 2401
502 = 2500

## Properties

The number m is a square number if and only if one can arrange m points in a square:

 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25

The formula for the nth square number is n2. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Four is the smallest Composite number, its proper Divisors being and. In mathematics Nine is a Composite number, its proper Divisors being 1 and 3. 25 ( twenty-five) is the Natural number following 24 and preceding 26. 36 ( thirty-six) is the Natural number following 35 and preceding 37. This page is for the number For the steamboat see Forty-Nine (steamboat 49 ( forty-nine) is the Natural number following 48 64 ( sixty-four) is the Natural number following 63 and preceding 65. 81 ( eighty-one) is the Natural number following 80 and preceding 82. 196 is a Natural number following 195 and preceding 197. It is the square of 14. This is also equal to the sum of the first n odd numbers

$n^2 = \sum_{k=1}^n(2k-1)$

as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (marked as '+'). In Mathematics, the parity of an object states whether it is even or odd So for example, 52 = 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 n2 = 2(n − 1)2 − (n − 2)2 + 2. For example, 2×52 − 42 + 2 = 2×25 − 16 + 2 = 50 − 16 + 2 = 36 = 62.

A square number is also the sum of two consecutive triangular numbers. A triangular number is the sum of the n Natural numbers from 1 to n. The sum of two consecutive square numbers is a centered square number. In Elementary number theory, a centered square number is a centered Figurate number that gives the number of dots in a square with a dot in the Every odd square is also a centered octagonal number. A centered octagonal number is a centered Figurate number that represents an Octagon with a dot in the center and all other dots surrounding the center

Lagrange's four-square theorem states that any positive integer can be written as the sum of 4 or fewer perfect squares. Lagrange's four-square theorem, also known as Bachet's conjecture, was proven in 1770 by Joseph Louis Lagrange. 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 Number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every Natural number k there exists an associated positive

A square number can only end with digits 00,1,4,6,9, or 25 in base 10, as follows:

1. If the last digit of a number is 0, its square ends in 00 and the preceding digits must also form a square. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.
2. If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.
3. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
4. If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four.
5. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
6. If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.

An easy way to find square numbers is to find two numbers which have a mean of it, 212:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22×20 = 440 + 12 = 441. This works because of the identity

(x − y)(x + y) = x2 − y2

known as the difference of two squares. In Mathematics, the difference of two squares is when a number is squared, or multiplied by itself and is then subtracted from another squared number Thus (21–1)(21 + 1) = 212 − 12 = 440, if you work backwards.

A square number cannot be a perfect number. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding

## Odd and even square numbers

Squares of even numbers are even, 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.

## Chen's theorem

Chen Jingrun showed in 1975 that there always exists a number P which is either a prime or product of two primes between n2 and (n + 1)2. Chen Jingrun ( May 22 1933 – March 19 1996) was a Chinese Mathematician who made significant contributions to Number In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a semiprime (also called biprime or 2- Almost prime, or pq number) is a Natural number that is the product See also Legendre's conjecture. Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a Prime number between n 2 and ( n  + 12