A perfect number is a positive integer which equals the sum
of all its divisors, excluding itself. The first perfect number is 6 since 6 =
1 + 2 + 3. The next perfect number is 28 since 28 = 1 + 2 + 4 + 7 + 14.
Over two thousand years ago, Euclid found a formula for
generating perfect numbers. The product of
(2p-1) and (2p-1) is a perfect number when p is a prime and
2p-1 is also a prime number. Many high prime numbers can be
found using the formula 2p-1, but the formula doesn’t always produce
prime numbers. When the formula does produce a prime number, it is called a Mersenne
prime named after the French Friar, Marin Mersenne (Marin Mersenne - Wikipedia).
The ancients knew of the first four perfect numbers: 6, 28, 496
and 8128. Through trial division, the list was confirmed with three additional
perfect numbers: 33550336, 8589869056 and 137438691328. In 1772 Euler found the eighth:
2305843008139952128 (with no modern computing aids). Euler also proved the
converse of Euclid’s original proof, that is even numbers are perfect if and
only if they can be expressed in the form (2p-1) (2p-1).
It is known that there are an infinite number of primes;
however, it is not known if there are an infinite number of Mersenne prime
numbers. As of this posting, there have been 51 confirmed Mersenne primes (the
largest has over 24 million digits). If one were able to prove the Mersenne
primes continue infinitely, then so would perfect numbers. The size of the
perfect number associated with the largest known Mersenne prime has over 49
million digits.
Here is a link to a spreadsheet
for calculating the first eight perfect numbers.
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