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**Amicable numbers **    Amicable numbers are two integer numbers if one is the sum of the proper positives divisors of the other one, the divisor major than zero and minor than the number, and vice versa, this is, if //‘a’ // and //‘b’ // are amicable numbers; //‘b’ // is the sum of the proper positives divisors of //‘a’ // and //‘a’ // is the sum of the proper positives divisors of //‘b’ //<span style="color: #000000; font-family: Arial,Helvetica,sans-serif;">. For example: If we search the positives divisors of 220 we get: <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;"> 1x220 = 2x110 = 4x55 = 5x44 = 10x22 = 11x20 If we extract the proper divisors and we add that: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 If we search the positives divisors of 284 we get: <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;"> 1 x 284 = 2x142 = 4x71 If we extract the proper divisors and we add that: 1 + 2 + 4 + 71 + 142 = 220 Then 220 and 248 are a pair amicable numbers, knew by the Pythagoreans. <span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">The Pythagoreans thought that these amicable numbers, 220 and 284 has mystical properties. The people thought that they should make two persons become friends using those properties, giving them food whit charasteristics of two amicables numbers in different place at the same time. Some amicable numbers are easy to find with the formula discovered by Leonhard Euler (1707-1783) in the year 1750, which is absurdly attributed to Thabit ibn Qurra (826-901), sometimes to René Descartes (1596-1650), or Pierre de Fermat (1601-1665), whom discovered three pairs of amicable numbers ( <span style="color: #d99594; font-family: Arial,Helvetica,sans-serif;">6232, 6368 <span style="font-family: Arial,Helvetica,sans-serif;">; <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;">9.363.584, 9.437.056 <span style="color: #d99594; font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">(1638); <span style="color: #d99594; font-family: Arial,Helvetica,sans-serif;"> 17.296, 18.416 <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">(1636)) correspondently. The formula is: If // p = //3 × 2// n //-1 1, // q = // 3 × 2 // n // <span style="font-family: Arial,Helvetica,sans-serif;"> 1, // r // = 9 × 22 // n // -1 <span style="font-family: Arial,Helvetica,sans-serif;"> 1 <span style="font-family: Arial,Helvetica,sans-serif;">, where // n //<span style="font-family: Arial,Helvetica,sans-serif;"> > 1 <span style="font-family: Arial,Helvetica,sans-serif;">is an integer and // p //, // q // , <span style="font-family: Arial,Helvetica,sans-serif;">and // r //<span style="font-family: Arial,Helvetica,sans-serif;"> are prime numbers, then 2 // npq // <span style="font-family: Arial,Helvetica,sans-serif;">and 2 // nr //<span style="font-family: Arial,Helvetica,sans-serif;"> are a pair of amicable numbers. Euler discovered 60 pairs of amicable numbers with his formula in 1750. But the second pair ( <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;">1184, 1210 <span style="font-family: Arial,Helvetica,sans-serif;">) was ignored by the scientists and was discovered in 1866 by Niccolò Paganini, an Italian boy with 16 years old. The pair <span style="color: #d99594; font-family: Arial,Helvetica,sans-serif;">6232, 6368 <span style="font-family: Arial,Helvetica,sans-serif;"> can not be found with the formula. <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif;">
 * Definition **
 * <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;">220 **<span style="color: #000000; font-family: Arial,Helvetica,sans-serif;">
 * <span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;">284 **<span style="color: #92cddc; font-family: Arial,Helvetica,sans-serif;"> <span style="color: #000000; font-family: Arial,Helvetica,sans-serif;">
 * Description **



** Bibliography: ** <span style="font-family: Arial,Helvetica,sans-serif;">*[] guyyyyyyyyy !!!! =-)!!!!
 * “Matemática divertida y curiosa ” by Malba Tahan. 2007, Intermedio Ediciones Ltda. Bogotá, Colombia. Pp. 23, 24 and 65.han. 2007, Intermedio Ediciones Ltda. Bogotá Colombia. Pp. 23, 24 and 65.
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