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A positive integer that has no perfect square Divisor s 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 ''n''th square number is ''n''2. This is also equal to the sum of the first ''n'' Odd Number s (), 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 ''n''th 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 (). 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 ''4''2 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'' = ''50''2 + ''50'' + ''51'' + ''51'' + ''52'' = ''2500'' + ''204'' = ''2704''. A square number is also the sum of two consecutive Triangular Number s. 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. 3 squares are not sufficient for numbers of the form 4''k''(8''m'' + 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 4''k''+3. This is generalized by Waring's Problem . A square number can only end with digit 0,1,4,5,6,9, following these rules: #If the last digit of a number is 1 or 9, its square ends in 1 and the preceding Digit must be even. #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 preceding digit must also be even. #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 is 5 or 0, its square ends in the last digit of the root and the preceding digits must be square. 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: 22x20=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't be a Perfect Number . EXTERNAL LINKS
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