45th and 46th Mersenne Primes Confirmed

kahunak writes to alert us that GIMPS has announced that the 45th and 46th Mersenne primes have been confirmed. The EFF's $100,000 award, for the first prime over 10 million digits in length, will probably be claimed. (We discussed no. 45 when it was announced.)

Re:Prime Post!

[QuickFox]

Not quite. In fact I will hereby reveal to the world the exact beginning and the exact ending of the 47th Mersenne prime (not just the 45th or the 46th, really the 47th!) as written in binary notation.

Not kidding, dead serious, this is the real thing:

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 ... ... ... 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

Mersenne numbers are by definition 2^n-1, which means that in binary notation every such number is a sequence of ones.

This doesn't matter so much

[l2718]

I think this tell us a lot more about the potential power of distributed computing than about prime numbers. While Mersenne primes are interesting to number theorists, we'll never find enough to do statistics on -- they are mostly of interests to pure mathematicians for reasons of curiosity. Random prime numbers of about 1024 bits are much more useful (and easier to find). On the other hand, if these was ever a problem we really needed to solve (protein-folding screensavers come to mind) then we now know how much computation power we can harness.

Re:Prime Post!

[Mprx]

Infinity of Mersenne primes isn't proven, you'll have to post the whole thing for us to believe you.

Re:So

[Thail]

You do realize that sending this prime number via SMS would require 62501 sms messages, using the standard US sms character limit of 160. Let's see... at the standard rate for SMS in the US of $.20, ( http://tech.yahoo.com/blogs/patterson/26695/congress-to-cell-carriers-why-have-sms-rates-doubled/ [yahoo.com] ) that comes out to, $12,500.20.

Now, assuming you can SMS at lightning speed and input 3 characters per second on a non qwerty keyboard (which is pretty dang fast if this story is to be believed http://www.engadget.com/2004/11/17/new-world-record-for-fastest-text-messaging/ [engadget.com] ) typing that out will take roughly 926 hours or 38.5 days.

Now I'm not a doctor, but you'd also have to factor in the chance for physical, and mental harm from this extended bout of texting. No sleep, no food or water, and definitly no slashdot for 38.5 days, not to mention the incedible amount of stress placed upon the joints, tendons, and muscles of your thumbs and arms.

I say no thank you sir, no thank you indeed. Good luck in your epic endeavor!

Re:This doesn't matter so much

[kevinatilusa]

Knowledge of whether or not there are infinitely many Mersenne primes would probably not be interesting even to most pure mathematicians -- it's sort of a bizarre question that seems disconnected from the rest of mathematics. What would be interesting would be the actual methods used to prove this. In practice almost every question involving the existence/non-existence of certain types of primes is one we already know the answer to.

The reason for this lies in the prime number theorem, which says that the proportion of numbers less than N which are prime is about 1/Log(N). Unless there's some compelling reason to believe otherwise, you can guess the answer to many problems involving primes by replacing them with a set randomly chosen with the same probability.

For example, a randomly chosen number near 2^p-1 will be prime with probability about proportional to 1/p. Since the sum of 1/p diverges, we expect there to be infinitely many Mersenne primes (and can even guess their number, though this requires a bit more careful analysis to take care of the observation that Mersenne numbers don't have small prime factors, but this should only increase their number).

The same trick allows us to guess the answer for twin primes (sum diverges, so there should be infinitely many) and Fermat primes (primes of the form 2^(2^n)+1 -- the sum converges, so there should be only finitely many). But none of these are really rigorous proofs, because they're all based on the fundamental assumption that the primes are somehow pseudorandom.

Depending on the method of attack, a proof of the infinitude of Mersenne Primes may also shed light on how accurate or inaccurate the pseudorandomness assumption is. I would consider that to be a VERY interesting question.

Re:Prime Post!

[grimJester]

Mersenne numbers are by definition 2^n-1, which means that in binary notation every such number is a sequence of ones.

Since the length of the number in base-10 is a little more than 10 million digits, and 10^3 is roughly 2^10, does this mean n is as low as 33-35 million?

Nevermind, checked Wikipedia. The largest currently known n is 32,582,657 so apparently my reasoning is correct.