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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.
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 square numbers up to and including m.
- 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
The pattern between any perfect square from negative infinity to positive infinity is as follows,
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.
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.
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 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.
A square number can only end with digits 00,1,4,6,9, or 25 in base 10, as follows:
- If the last digit of a number is 0, its square ends in 00 and the preceding digits must also form a square.
- 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.
- If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
- 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.
- If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
- 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
The formula to calculate any square number is: (n1)2 + (dn1) + (dn2)
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 and 440 + 12 = 441. This works because of the identity
known as the difference of two squares. Thus (21 − 1)(21 + 1) = 212 − 12 = 440, if you work backwards.
- 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.
- Integer square root
- Methods of computing square roots
- Quadratic residue
- Polygonal number
- Triangular square number
- Euler's four-square identity
- Automorphic number
- Sum of odd numbers
- Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30-32, 1996. ISBN 0-387-97993-X
- 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.
- Knitted wall-hanging showing square numbers as the sum of odd numbers