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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, 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 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 Properties The number m is a square number if and only if one can arrange m points in a square: 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 adding the (n − 1)th square to itself, subtracting the n2th square number, and adding 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 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: 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. Chens 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 See also | ||||||||
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