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

Properties

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

12 = 1

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22 = 4

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32 = 9

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42 = 16

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52 = 25

The formula for the nth square number is n². 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 '+'). 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. 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.

A square number cannot be a perfect number.

Easy ways to calculate square numbers

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.

special cases

• If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m*(m+1) and represents digits before 25. For example the square of 65 can be calculated by 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 n = m². For example the square of 70 is 4900.

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.

See also

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

References

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

• Learn Square Numbers Practice square numbers up to 144 with this children's multiplication game
• 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
• The first 1,000,000 perfect squares Includes a program for generating perfect squares up to 10^15.