Show that no proper divisor of a perfect number is perfect.

proof. let n be perfect, and let be divisors of n. Now, .

Now, what should I do to get O(d) not equal to 2d for all ds?

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- Mar 18th 2008, 12:07 PMtttcomraderDivisor of perfect number is not perfect
Show that no proper divisor of a perfect number is perfect.

proof. let n be perfect, and let be divisors of n. Now, .

Now, what should I do to get O(d) not equal to 2d for all ds? - Mar 18th 2008, 02:26 PMOpalg
Try proving the contrapositive: if d is perfect and n=kd is a multiple of d, then n is not perfect. To see this, notice that if d is the sum of its proper divisors, say , then . So n is the sum of some of its proper divisors. But n also has other divisors that are not multiples of k (for example, 1). Therefore the sum of the proper divisors of n is greater than n.