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What are amicable numbers? ... PDF Print E-mail
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Friday, 27 March 2009 13:21

What are amicable numbers? ...

 

In Number Theory, the part of mathematics that regards the numbers and their properties, play an important role the divisors of numbers, and their sum.

 

And therefore, we can define,

Definition 1. A number is called perfect,  is σ(n) = 2n.

Examples.

28 is a perfect number. In fact, the divisors of 28 are: 1+2+4+7+14+28=56.

6 is a perfect number. In fact, the divisors of 6 are: 1+2+3+6=12.

Theorem. All the even perfect numbers have the form: 2^(n-1)x(2^n - 1).

Open problem. There are no perfect odd numbers. 

Definition 2. Two numbers are amicable, if σ(a) = b and σ(b) = a.

Examples.

1184 and 1210 are amicable. 

In fact,  

1+2+4+8+16+32+37+74+148+296+592=1210  and 

1+2+5+10+11+22+55+110+121+242+605=1184

Amicable numbers generally may be created in chain form, like in the figure above.

Example.

2620, 2924

5020, 5564

6232, 6368

1114488537712, 1188853793424 

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