It has now been confirmed that two new Mersenne primes have been found-and hence, we now have two new perfect numbers. They are the 45th and 46th known perfect numbers. The ancient Greeks only knew of four.
Here is the press release: <http://www.mersenne.org/m45and46.htm>http://www.mersenne.org/m45and46.htm
Check the Mersenne prime website for more information and updates: <http://www.mersenne.org/>http://www.mersenne.org/ (There is more technical and historical information at <http://primes.utm.edu/mersenne/index.html>http://primes.utm.edu/mersenne/index.html though, at last check, that page had not yet been updated to show the 45th and 46th perfect numbers)
The newly found Mersenne primes can be expressed as 243,112,609 -1 and 237,156,667 -1 (following the standard Mersenne pattern of 2p - 1, where p is prime). The corresponding perfect numbers are 2p-1 (2p - 1 ). The two Mersenne primes have 12,978,189 and 11,185,272 digits, so the two perfect numbers have 25,974,378 and 22,370,544 digits. (I calculate that a page of Arial type with font size 10 [the type of this email] has 84 characters per line and 56 lines or 4704 characters per page. At that rate, the longer new perfect number would need 5522 pages to be printed; the 25+ million characters, written out in one line of Arial 10 characters would be 51 km [about 32 miles] long).
Recall that a perfect number is a number for which the sum of all its proper factors is itself. For example 6 = 1 +2 + 3 is the first, 28 = 1 + 2 + 4 + 7 + 14 is the second, 496, 8128, and 33,550,336 are the third, fourth, and fifth (check them yourself!). All even perfect numbers are of the form 2p-1 (2p - 1 ), where p is prime and also the part in square brackets is prime. The part in square brackets is called a Mersenne prime. Hence, whenever a new Mersenne prime is found, a new perfect number follows. (incidentally, no one has ever found an odd perfect number, but no one has proven that they cannot exist)
Now we are awaiting the 47th !
--Akcooper 17:23, 23 September 2008 (UTC)
Does anyone know if this found with a supercomputer or by distributing the processing power over a lot of PCs (like folding @ home)?
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The information I found on Mersenne primes is a little different than the article that was pasted here. When I looked on websites (including mersenne.org), the following was the explanation given:
A Mersenne prime is a Mersenne number, i.e., a number of the form $ M_n=2^n-1 $ that is prime. The first Mersenne primes are 3, 7, 31, 127 (corresponding to n = 2, 3, 5, 7).
--Bpitstic
