It has been suggested that this article or section be merged into Square (algebra). (Discuss) In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an 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 square numbers up to and including m. Contents 1 Examples 2 Properties 3 Special cases 4 Odd and even square numbers 5 See also 6 References 7 Further reading 8 External links Examples The squares (sequence A000290 in OEIS) below 502 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 The difference between any perfect square and its predecessor is given by the following identity,

Also, it is possible to count up square numbers by adding together the last square, the last square's root, and the current root. Mathematically expressed, n2 = n − 1]2 + n − 1] + n. Properties The number m is a square number if and only if one can arrange m points in a square: m = 12 = 1 m = 22 = 4 m = 32 = 9 m = 42 = 16 m = 52 = 25 The expression 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 (shown in cyan). The formula follows: 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. Another property of a square number is that the number of its divisor is odd, while other number have even number of divisors. An integer root is the only divisor that pairs up with itself to yield the number, while other divisors come in pairs. 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 end only 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. In base 16, a square number can end only with 0,1,4 or 9 and - in case 0, only 0,1,4,9 can precede it, - in case 4, only even numbers can precede it.

There is a simple Boolean formula to perform this test on any number n using only the least significant byte: if (n & 7 == 1) || (n & 31 == 4) or (n & 127 == 16) or (n & 191 == 0) then return n is probably square else return n is definitely not square In general, if a prime divides a number then also the square of that prime must divide the number, if it fails to divide it second time, the number is definitely not square. Every prime must divide the number even times. At certain point it is faster to run square root on the number than to test it for all primes up to 4th root of the number. Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, just test for squarity: for given m and some number k, if k²- m is square of any number n then k - n divides m. For example 100² - 9991 is square of 3, consequently 100 - 3 divides 9991. This test is deterministic for odd divisors in range from k - n to k + n where k covers some range of natural numbers k ≥ √m. A square number cannot be a perfect number.

Square and Triangle numbers can be converted both ways using Heath's Squareangular theorm (as invented by Oscar Heath Stephens) it goes as follows. T=triangle S=square number N=√S or Triangular root of T (S-N)/2+N=T or (T-N)2+N=S 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. If the number has two digits and is of the form 5m where m represents the units digit, its square is AABB where AA = 25 + m and BB = m2. Example: To calculate the square of 57, 25 + 7 = 32 and 72 = 49, which means 572 = 3249. 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. See also Brahmagupta–Fibonacci identity The Book of Squares Integer square root Methods of computing square roots Quadratic residue Polygonal number Square triangular number Euler's four-square identity Automorphic number Power of two References Weisstein, Eric W., "Square Number" from MathWorld. 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.