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, a number whose square root is an integer.) So for example, 9 is a square number since it can be written as 3 × 3. If rational numbers are included, then the ratio of two square integers is also a square (e.g. 4/9 = 2/3 × 2/3).

A positive integer that has no perfect square divisors except 1 is called square-free.

Examples

The first 51 squares (sequence A000290 in the OEIS) are: 0² = 0

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

Bold textLink titleItalic text--81.145.240.195 22:25, 5 February 2007 (UTC)Strike-through text==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 ($n^{2}=\sum _{k=1}^{n}(2k-1)$), 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.

Headline text

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 ($n^{2}=2(n-1)^{2}-(n-2)^{2}+2$). 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.

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 can 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

• http://www.alpertron.com.ar/FSQUARES.HTM is a Java applet that decomposes a natural number into a sum of up to four squares.

See also

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