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. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... The integers are commonly denoted by the above symbol. ... In algebra, the square of a number is that number multiplied by itself. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

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 leaving a remainder. ... In mathematics, a square-free integer is one divisible by no perfect square, except 1. ...

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". 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). “Exponent” redirects here. ...

Examples

The first 51 squares (sequence A000290 in OEIS) are: 0² = 0 The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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. 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 (marked as '+'). So for example, 5² = 25 = 1 + 3 + 5 + 7 + 9. 1 as a square number. ... 4 as a square number. ... 9 as a square number. ... 16 as a square number. ... Image File history File links Square_number_25. ... In mathematics, any integer (whole number) is either even or odd. ...

The nth square number can be calculated from the previous two by adding the (n − 1)th square to itself, subtracting the (n − 2)th square number, and adding 2 (n² = 2(n − 1)² − (n − 2)² + 2). For example, 2×5² − 4² + 2 = 2×25 − 16 + 2 = 50 − 16 + 2 = 36 = 6².

It is often also useful to note that the square of any number can be represented as the sum 1 + 1 + 2 + 2 + ... + n − 1 + n − 1 + n. For instance, the square of 4 or 4² is equal to 1 + 1 + 2 + 2 + 3 + 3 + 4 = 16. This is the result of adding a column and row of thickness 1 to the square graph of three (like a tic tac toe board). You add three to the side and four to the top to get four squared. This can also be useful for finding the square of a big number quickly. For instance, the square of 52 = 50² + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.

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. A triangular number is the sum of the n natural numbers from 1 to n. ... A centered square number is a centered figurate number that represents a square with a dot in the center and all other dots surrounding the center up to a certain city block distance, i. ... 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 dot in successive octagonal layers. ...

Lagrange's four-square 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. Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In mathematics, the integer prime-factorization (also known as prime decomposition) problem is this: given a positive integer, write it as a product of prime numbers. ... In number theory, Warings problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. ...

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.
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 For other uses, see Number (disambiguation). ... Digit may refer to: A finger or a toe Numerical digit, as used in mathematics or computer science Digit (unit), an ancient meterological unit Digit (magazine), an Indian information technology magazine This is a disambiguation page: a list of articles associated with the same title. ...

(x − y)(x + y) = x² − y²

known as the difference of two squares. Thus (21–1)(21 + 1) = 21² − 1² = 440, if you work backwards. In mathematics, the difference of two squares refers to the identity a2 − b2 = (a + b)(a − b) from elementary algebra. ...

A square number cannot be a perfect number. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...

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. See also Legendre's conjecture. This is a Chinese name; the family name is Chen Chen Jingrun (Traditional Chinese: ; Simplified Chinese: ; Hanyu Pinyin: Chén Jǐngrùn; Wade-Giles: Chen Chingjun, May 22, 1933–March 19, 1996) was a Chinese mathematician who made significant contributions to number theory. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ... Adrien-Marie Legendre conjectured that there is a prime number between n² and (n+1)² for every integer n > 0. ...

Further reading

• Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30-32, 1996. ISBN 0-387-97993-X

External links

• Dario Alpern, Sum of squares. A Java applet to decompose a natural number into a sum of up to four squares.
• Fibonacci and Square Numbers at Convergence

See also

• Integer square root
• Methods of computing square roots
• Quadratic residue
• Polygonal number
• triangular square number
• Euler's four-square identity
• Automorphic number