(Mostly) Confirmed: New Mersenne Prime Found

A reader writes "Distributed computing seems once more to be succesful. The combined effort of many pc's joining Primenet in search for a new Mersenne prime may have found there fifth result. Among them many belonging to /. readers. There is an unconfirmed claim for Mersenne prime #39 of over 3,500,000 digits, for which a considerable amount of money has been awarded. SETI looks for ET's messages, but found none sofar. Mersenne primes are used to tell ET about us. A previous found Mersenne number was used to show the advance of science on our planet in a message send into outer space. " The Primenet list has confirmed that while they still need to totally test it out (which should be done by the 24th), they believe that the number found today is the 39th positive.

Comments

F SETI@Home

[joshamania]

You can help cure cancer with http://foldingathome.standford.edu.

what a Mersenne prime is

[vrmlknight]

ok i had to look it up a smiple search on google dont think that this as a troll just trying to help those that dont use google
http://www.utm.edu/research/primes/mersenne.shtm l:

"Many early writers felt that the numbers of the form 2n-1 were prime for all primes n, but in 1536 Hudalricus Regius showed that 211-1 = 2047 was not prime (it is 23.89). By 1603 Pietro Cataldi had correctly verified that 217-1 and 219-1 were both prime, but then incorrectly stated 2n-1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct.
Enter French monk Marin Mersenne (1588-1648). Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2n-1 were prime for

n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257
and were composite for all other positive integers n 257. Mersenne's (incorrect) conjecture fared only slightly better than Regius', but still got his name attached to these numbers.
Definition: When 2n-1 is prime it is said to be a Mersenne prime.
It was obvious to Mersenne's peers that he could not have tested all of these numbers (in fact he admitted as much), but they could not test them either. It was not until over 100 years later, in 1750, that Euler verified the next number on Mersenne's and Regius' lists, 231-1, was prime. After another century, in 1876, Lucas verified 2127-1 was also prime. Seven years later Pervouchine showed 261-1 was prime, so Mersenne had missed this one. In the early 1900's Powers showed that Mersenne had also missed the primes 289-1 and 2107-1. Finally, by 1947 Mersenne's range, n 258, had been completely checked and it was determined that the correct list is:
n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

Many ancient cultures were concerned with the relationship of a number with the sum of its divisors, often giving mystic interpretations. Here we are concerned only with one such relationship:
Definition: A positive integer n is called a perfect number if it is equal to the sum of all of its positive divisors, excluding n itself.
For example, 6 is the first perfect number because 6=1+2+3. The next is 28=1+2+4+7+14. The next two are 496 and 8128. These four were all known before the time of Christ. Look at these numbers in the following partially factored form:
2.3, 4.7, 16.31, 64.127.
Do you notice they all have the same form 2n-1(2n-1) (for n = 2, 3, 5, and 7 respectively)? And that in each case 2n-1 was a Mersenne prime? In fact it is easy to show the following theorems:
Theorem One: k is an even perfect number if and only if it has the form 2n-1(2n-1) and 2n-1 is prime. [Proof.]
Theorem Two: If 2n-1 is prime, then so is n. [Proof.]

So the search for Mersennes is also the search for even perfect numbers!
You may have also noticed that the perfect numbers listed above (6, 28, 496, 8128) all end with either the digit 6 or the digit 8--this is also very easy to prove (but no, they do not continue to alternate 6, 8, 6, 8,...). If you like that digit pattern, look at the first four perfect numbers in binary:

110
11100
111110000
1111111000000
(The binary digit pattern is a consequence of Theorem One.) It is not known whether or not there is an odd perfect number, but if there is one it is big! This is probably the oldest unsolved problem in all of mathematics.
When checking to see if a Mersenne number is prime, we usually first look for any small divisors. The following theorem of Euler and Fermat is very useful in this regard.

Theorem Three: Let p and q be primes. If q divides Mp = 2p-1, then
q = +/-1 (mod 8) and q = 2kp + 1
for some integer k. [Proof.]
Finally, we offer the following for your perusal:
Theorem Four. Let p = 3 (mod 4) be prime. 2p+1 is also prime if and only if 2p+1 divides Mp. [Proof].
Theorem Five. If you sum the digits of any even perfect number (except 6), then sum the digits of the resulting number, and repeat this process until you get a single digit, that digit will be one "