Square number

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. A positive integer that has no perfect square divisors except 1 is called square-free. 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). There are <math> \lfloor \sqrt{n} \rfloor</math> square numbers up to and including n.

Contents 1 Examples 2 Properties 3 Odd and even square numbers 4 Chen's theorem 5 Further reading 6 External links 7 See also

Examples

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

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 (<math>n^2 = \sum_{k=1}^n(2k-1)</math>), 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. 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 (<math>n^2 = 2(n-1)^2-(n-2)^2+2</math>). 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. 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. 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

(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. A square number cannot be a perfect 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.

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