Perfect+Perfect=Perfect ?

How many perfect numbers can be expressed as the sum of two perfect numbers greater than 6?


The answer is 0.

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1 solution

Kalpok Guha
Apr 2, 2015

Let N 1 N_1 and N 2 N_2 are two perfect numbers greater than 6 6

When consider modulo 6 6 implies we consider modulo 2 2 and 3 3 Obviously modulo 2 2 gives 0 0 . And with p p as odd, modulo 3 3 gives ( 1 ) even × ( ( 1 ) odd 1 ) 1 (-1)^{\text{even}} \times ( (-1)^{\text{odd}} - 1 ) \equiv 1 . Which gives 4 4 modulo 6 6 .

So N 1 4 ( m o d 6 ) N_1 \equiv 4\pmod 6 and N 2 4 ( m o d 6 ) N_2 \equiv 4\pmod 6

Thus N 1 + N 2 2 ( m o d 6 ) N_1+N_2 \equiv 2\pmod 6 .

Thus N 1 + N 2 N_1+N_2 is not perfect.

So the answer is 0 \boxed{0}

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