One small beep brings prime number glory to Missouri

[T_i_B]

I have a question here. Prize money aside, what actual use is the world's largest prime number? I'm curious here as when I was 12 they broke this record and my maths teacher told my class that the world's largest prime number was of no use to anyone.

http://www.guardian.co.uk/usa/story/0,12271,1678100,00.html

If it takes 700 computers nine years to find the answer, it must be one beast of a question. But for researchers in Missouri, on a quest to find the world's largest prime number, it was all worthwhile.

In mid-December, a reassuring beep signalled the end of the long search, and one of the computers came up with a prime number with the mysterious name of M30402457. Made up of 9.1m digits, it trounces all others discovered so far. The number discovered in Missouri falls into a special category called mersenne prime numbers. These are expressed as the number 2 raised to the power of "p" minus one, where "p" is also a prime number.

For Steven Boone and Curtis Cooper at Central Missouri State University, it was a moment to remember. "It's a huge achievement, we're really excited," said Dr Boone. "People ask why we do this. It's like going on a quest. We're looking for something incredibly rare," he said. "It's the icing on the cake."

And then there is the financial incentive. The person who finds the first mersenne prime number more than 10m digits long stands to win $100,000 (£58,000), the prize of the Great Internet Mersenne Prime Search competition.

[htuttle]

1. Prime numbers are primarily used in encryption applications

...though I'm not sure that any existing encryption scheme uses anything but a fraction of the highest prime number.

Not sure it's used for anything else outside the realm of pure mathematics.

[benburch]

3. I use the idle time on my servers to run this application...

I know it has little practical value, but I cannot stand seeing all those compute cycles wasted.

[Deep13]

6. SETI has a similar thing ...

... where home comuter owners volunteer non-use time to process SETI data.

[benburch]

9. Yes, I was a beta-tester for that back in like 99.

I still run that on one or two of the slower machines.

White Rose involves NINE different machines...

[Deep13]

2. "more than 10m digits long "

Is that 10 million (1000 x 1000) or 10 thousand?

[Bill McBlueState]

10. 10 million digits

As far as I can tell, the number they found is (2^30402457)-1 . At least, that's what I assume the notation "M30402457" means.

That number would have nearly 10 million digits, since it's about equal to 10^9152051. That's still way smaller than a googolplex.

[Atman]

4. We'll need it to calculate the nation debt Bush is ringing up.

}(

[Deep13]

7. He shoots! He scores!

:applause:

[dweller]

5. thank you for not posting

the digits...

i'll take your word for it.
dp

[phantom power]

8. They may be useful for validating...

postulates about the distribution of primes among the integers.

One observation: what these guys are finding are Mersenne primes, which are just a subset of the primes. All of the largest known prime numbers are Mersenne primes, thanks to this project. Knowing a subset of primes might serve as useful land-marks for clever algorithms on very large numbers. Factoring mega-digit numbers, for example.

[T_i_B]

11. You lost me at integers

Is there anyway to explain all this in layman's terms? Or any pratical uses for any of this?

Sorry

[phantom power]

12. I'll take a stab...

I'm mostly a dilletante at number theory, but here's a start. Maybe a real number theory expert can step in :-)

The primes are sort of elemental among the integers. A prime integer is divisible by only one and itself. For example, the first few primes are {2, 3, 5, 7, 11, 13...}. The number 6 is not prime. It's "prime factorization" is (2)(3). The prime-factorization of 12 is (2)(2)(3).

There are an infinite number of primes. There is no "largest" prime, although there is a "largest known prime." There is no known "formula" for generating the primes. It's conjectured that there cannot be any such formula, but to my knowledge nobody can prove that. However, there are theorems regarding their density, as we go out towards infinity. They get more and more sparse, as we travel along the number line. The theorems for prime density are only "asymtotic," that is to say that they only become accurate as we approach infinity. In fact, the density of primes even for very large numbers is higher than the theorem's limit. So, finding primes far out on the number line can be useful for generating "actual" density statistics.

The primes have many uses, most of them flowing from the fact that a set of numbers {0, 1, ... Q}, where Q is prime, is a "field" over modular multiplication and addition, where the modulus is Q. It would take me a bit to explain what a field is, but essentially it's a set of numbers (or objects), together with two operations that are addition or multiplication (or that "act like" those two ops), which all "behave like" the numbers we're familiar with: they have a "zero" (additive identity), "1" (multiplicative identy), every object (except zero) has a multiplicative inverse, i.e. "1/x" is defined.

[T_i_B]

13. I know what a prime number is BUT........

......I have no idea what an integer is. I might be able to understand if you explain in gumby terms. Otherwise I get confused. :-)

[phantom power]

14. Oh, the integers are just the numbers {... -2, -1, 0, 1, 2, ...}

No fractions (rationals), decimals (reals), etc.

[eppur_se_muova]

15. It's not just thanks to this project...it's in the nature of Mersennes.

There are particular rules which apply to both trial factoring and primality proofs of Mersenne primes that do not apply to primes in general. It is quite likely that the largest known prime number at any given time will always be a Mersenne prime, and it was only for a few brief months (IIRC) a number of years ago that this was not true.

In short, Mersenne primes are *easier* to find than big primes of more general form. However, the study of Mersenne primes may lead to the discovery of more general principles which are more broadly applicable. Like most scientific research, scientists are just looking for something interesting, without necessarily having too much preference for what that might be.

Mersenne numbers are used in the convolution algorithms central to Apple's QuickTime compression technology, invented by Dr. Richard Crandall, who used to be a Chief Scientist at NeXT (Steve Jobs' startup after he got booted by Apple) and returned to Apple as an Apple Fellow, IIRC (memory a little wobbly). He is now an adjunct prof. at Reed College, OR,
http://www.google.com/u/Reed?q=Crandall&x=0&y=0 , and has his own software consulting company, Perfectly Scientific
http://www.perfsci.com/. He wrote the large-integer arithmetic algorithms which George Woltman has optimized in Prime95. (He's also a nice guy who answered my ignorant emails very helpfully!)