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WHAT IS THE THIRD PER

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JUD MCCRANIE

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Nov 30, 1995, 3:00:00 AM11/30/95
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On 29 Nov 95 09:23am, ERIK MOLEKAMP wrote to ALL:


EM> The program I wrote (in QuickBASIC 4.50) came up with 496 as the third
EM> perfect number.

...
EM> See a pattern yet? I found that 2^(n - 1) * (2^n - 1) is a perfect
EM> number for n integer and greater than 1, if (2^n -1) is a prime number.

This is all well-known. All even perfect numbers are of that
form, where 2^n-1 is a (Mersenne) prime.

EM> Questions left:

EM> Is it possible to either
EM> 1) find a perfect number that is not constructed in this way, or
EM> 2) prove that all perfect numbers are given by the above conditions?

EM> Can all odd numbers be ruled out? (proof)

All even perfect numbers are that way. It is still an open
question as to if there are any odd perfect numbers. The best
guess is probably that there aren't any odd perfect numbers.


Jud McCranie

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