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42nd Mersenne Prime Probably Discovered
RTKfan writes "Chalk up another achievement for distributed computing! MathWorld is reporting that the 42nd, and now-largest, Mersenne Prime has probably been discovered. The number in question is currently being double-checked by George Woltman, organizer of GIMPS (the Great Internet Mersenne Prime Search). If this pans out, GIMPS will have been responsible for the eight current largest Mersenne Primes ever discovered."
Can someone explain what the application/use these primes are for? Not a flame, I'm honestly curious as to what something like this could be used for, as are others, I'm sure.
Well, yeah, if you encode the Prime number in Binary it will not look Random at all. It will look like a giant string of 1s though... Aliens might mistake it for filler or something.
I read the internet for the articles.
...waste of time, money and processing power. what kind of use would this have, other than just knowing it? its like winning a eating contest: a completely useless achievement, plus it just turns to poop.
Yeah, except that the Mersenne primes are well known and thus useless for cryptography -- at least, if you want any security.
You are arguing about a different domain.
Encryption discussions have to take place in a "computing" domain, where a prime only exists if it has been computed to be prime by at least one computer somewhere in the world, and where the prime number can fit on a distribution medium.
Arguing that there are as many Mersenne primes as regular primes is only possible in a theoretical domain in which countably infinite sets can be said to exist.
The X axis is infinite. So are the Y and Z axes. Therefore, there must be an infinite number of regular solids. Oh, wait! There's only five. Gee, I guess the mere fact that numbers are infinite doesn't imply that subsets of those numbers are infinite.
"I'm not impatient. I just hate waiting." - My Dad
Actually, it doesn't have to be. It only suffices so say, that when you multiply all primes in your list, and add one, you get a number not divisable by any of the number in the list. Hence, one of TWO things can hold true:
1) The number in question really is prime, as you suggested
2) The number in question isn't prime. Then it has prime divisors, none of them in your list (because none of the primes in the list divided our new number).
In both cases, we have derived a way to find at least one new prime from any list of primes, and hence, the collection of primes is non-finite because we can always find "another one".
Yes, but we haven't actually proved that the result of the multiplication-addition operation is prime. However, we don't have to, because if it is prime, then it's necessarily a prime that's not on the list, and if it's composite, then by the Fundamental Theorem of Arithmetic it has a unique set of prime factors, and by the way you constructed the number, none of those primes are in the list. Either way, you've found at least one prime that wasn't in the list, contradicting the claim that the list was complete.
You see? For completeness, we say "either N is prime, in which case the list is incomplete, or N is composite, in which case the list is incomplete", and save ourselves worrying about whether N is prime or not.