Definition+&+Description+-+Version+1+with+classmate+comments



 ** Amicable numbers ** ** Definition ** 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’//. For example: If we search the positives divisors of 220 we get:  ** 220 **  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:  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.   ** Descriptiotion ** (just wrong written, change it for Description)  The Pythagoreans thought that these amicable numbers, 220 and 284 has mystical proprieties (just wrong written, should say properties) . The people thought that they could make that (this second that is not needed)  two persons become friends using those proprieties (just wrong written, should say properties) . <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;">(In the previous lines i found the talk about this description not very well developed, it says that people could make friend by the use of the properties but it's not quite explained like how they did it in a general way, feels like more is needed) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;">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="font-family: Arial,Helvetica,sans-serif;"> 6232, 6368 ; 9.363.584, 9.437.056 (1638); 17.296, 18.416  (1636)) correspondently. The formula is: If // p = // 3 × 2// n //-1 1, // q = //<span style="font-family: Arial,Helvetica,sans-serif;"> 3 × 2// n //  1, // r // = 9 × 22// n //-1  1, where  // n // > 1 is an integer and  // p // , // q //, and  // r // are prime numbers, then  2// n //// pq // and  2// n //// r //  are a pair of amicable numbers. Euler discovered 60 pairs of amicable numbers whit <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;">(wrong written: with) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;"> his formula in 1750. But the second pair ( <span style="font-family: Arial,Helvetica,sans-serif;"> 1184, 1210 ) was ignored by the scientists and it <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif;">(the it is not needed) <span style="font-family: Arial,Helvetica,sans-serif;"> was discovered in 1866 by Niccolò Paganini, an Italian boy with <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif;">(this with is not truly neccesary) <span style="font-family: Arial,Helvetica,sans-serif;"> 16 years old. The pair 6232, 6368 can not be found with the formula. (More than i said i can't find may problems on the draft, is very well written, with the correct usage of the characteristics of description and definition, congratulations, just try to correct the flaws and it will be done! Also like the subject! i didn't known about those pairs of numbers)
 * 284 ** <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;">

** Bibliography: ** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;">*http://www.todoexpertos.com/categorias/ciencias-e-ingenieria/matematicas/respuestas/617111/numeros-amigos
 * “Matemática divertida y curiosa ” by Malba Tahan. 2007, Intermedio Ediciones Ltda. Bogotá Colombia. Pp. 23, 24 and 65.
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