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### Perfect Numbers

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Date: 10/17/96 at 20:18:58
From: Anonymous
Subject: Perfect numbers

Is there any way, other than by trial and error, to figure out what
the perfect numbers are?
```

```
Date: 11/05/96
From: Doctor Yiu
Subject: Re: Perfect numbers

Dear Tahquitz,

A perfect number is a positive integer that is equal to the sum of
all its divisors, including 1, but not the number itself.

The two smallest perfect numbers are

6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14.

The search for perfect numbers began in ancient times. Euclid (3rd
Century BC) gave a construction of perfect number based on the notion
of prime numbers:

If (2^k-1) is a prime number, then 2^(k-1)(2^k-1) is a perfect number.

For example:

If k = 2, 2^k-1 = 3 is prime, which gives
2^(2-1)(2^2-1) = 2 times 3 = 6, the first (even) perfect number.

Similarly,

for k = 3,   2^k-1 = 7  gives  28,

for k = 5,   2^k-1 = 31  gives 2^4 x 31 = 496,  etc. etc.

(Note: When k = 4, (2^k-1) = 15 is NOT a prime number, and we CANNOT
generate a perfect number by this method).

Since the 17th Century, the number (2^k-1) has been known as a
Mersenne number, denoted by M_k.

In the 18th Century, the great Swiss mathematician Euler proved that
all EVEN perfect numbers must be of the form

2^(k-1)(2^k-1), where (2^k-1) is PRIME.

(It is not known whether or not there is an ODD perfect number.)

A new record for the largest KNOWN perfect number was obtained in
September 1996, along with a record for largest KNOWN Mersenne primes.
Before that there were 33 known Mersenne primes. D. Slowinski and
P. Gage also announced their discovery of a new Mersenne prime:

2^1257787 - 1, which has 378,632 digits.

This gives

2^1257786 x 2^1257787 - 1,

for a record largest KNOWN even perfect number.

You can find more about all of these on the Internet at Neal Calkin's
page:

http://www.utm.edu/research/primes/mersenne.shtml

-Doctor Yiu,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

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Associated Topics:
High School History/Biography
High School Number Theory
Middle School History/Biography

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