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My Prime Is Bigger Than Yours

K

KeyserXSoze

Gone But Not Forgotten
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http://story.news.yahoo.com/news?tmpl=story&cid=528&ncid=528&e=10&u=/ap/20031211/ap_on_hi_te/biggest_prime_number
Student Finds Largest Known Prime Number
Thu Dec 11, 6:12 AM ET
By JIM IRWIN, Associated Press Writer

DETROIT - More than 200,000 computers spent years looking for the largest known prime number. It turned up on Michigan State University graduate student Michael Shafer's off-the-shelf PC.

"It was just a matter of time," Shafer said.

The number is 6,320,430 digits long and would need 1,400 to 1,500 pages to write out. It is more than 2 million digits larger than the previous largest known prime number.

Shafer, 26, helped find the number as a volunteer on an eight-year-old project called the Great Internet Mersenne Prime Search.

Tens of thousands of people volunteered the use of their PCs in a worldwide project that harnessed the power of 211,000 computers, in effect creating a supercomputer capable of performing 9 trillion calculations per second. Participants could run the mathematical analysis program on their computers in the background, as they worked on other tasks.

Shafer ran an ordinary Dell computer in his office for 19 days until Nov. 17, when he glanced at the screen and saw "New Mersenne prime found."

A prime number is a positive number divisible only by itself and one: 2, 3, 5, 7 and so on. Mersenne primes are a special category, expressed as 2 to the "p" power minus 1, where "p" also is a prime number.

In the case of Shafer's discovery, it was 2 to the 20,996,011th power minus 1. The find was independently verified by other participants in the project.

Mersenne primes are rare but are critical to the branch of mathematics called number theory. That said, what is the practical significance of Shafer's number?

"People are going to make posters of it to hang up on the wall," said Shafer, who is pursuing a doctorate in chemical engineering. "It's a neat accomplishment, but it really doesn't have any applicability."

As for his own standing in the world of mathematics, "I don't think I'm going to be recognized as I go down the street or anything like that."

He said the method by which the number was found — harnessing many computers together — is more important than the number itself.

"Somebody else could have found the number," he said. "You install the program on the computer and it takes care of itself." But "I get the credit, along with the people that developed the software."
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Fascinating. I imagine what takes the time is not so much the finding of such a number, as the verification. Imagine how many potential integral factors there are for a number with over six million digits. I mean, are they absolutely certain it's prime? It must be a very powerful algorithm to check this.
Back in pre-Internet days, I was proud to know some reasonably large primes myself. The biggest one I can reel off from memory is 618,970,019,642,690,137,449,562,111. I do have a copy of a 24,000-digit Mersenne prime, which at the time was the world's largest yet found.
I guess they'll just keep on finding larger and larger ones as computer processing power increases.

Bill.
 
Huge new prime number discovered

Mathematicians in California could be in line for a $100,000 prize (£54,000) for finding a new prime number which has 13 million digits.

Prime numbers can be divided only by themselves and one.

The prize was set up by the Electronic Frontier Foundation to promote co-operative computing on the Internet.

The team from the University of California at Los Angeles (UCLA) found the new number by linking 75 computers and harnessing their unused power.

This enabled them to perform the enormous number of calculations needed to find and verify a new prime.

Thousands of people around the world linked the powers of their personal computers in the search for a higher "Mersenne" prime number - named after 17th-Century French mathematician Marin Mersenne.

Mersenne primes are expressed as two to the power of P, minus one - with P being itself a prime number.

Edson Smith, the leader of the winning UCLA team, told the Associated Press news agency: "We're delighted. Now we're looking for the next one, despite the odds."

http://news.bbc.co.uk/1/hi/world/americas/7640183.stm
 
Interesting piece here, but for some reason I can't copy the text. (This has happened to me before with some Telegraph articles.)


Prime number breakthrough
Virtually unknown professor takes major step towards solving a numerical problem which has baffled the finest minds for centuries.

http://www.telegraph.co.uk/science/scie ... essor.html
 
I enjoy memorising primes.

The biggest prime I know, strictly from memory, without looking it up, is 686479766013060971498190079908139321726943530014330540939446345918554318339765605212255964066145455497729631139148085803
7121987999716643812574028291115057151.

This is the Mersenne prime P=521 (in other words, you multiply 2 by itself 521 times, and subtract 1 from the answer.

Interestingly not all integers of the form 2 to the power P minus 1, where P is a prime, are themselves prime. Many are however. It is a great way to find monster primes.

There must exist, primes which are light years in length, when written in this size font. Fascinating, as Spock would say.

Bill.
Code:
 
JamesWhitehead said:
I'm impressed. Now can you solve the world's problems. Please. :)
Click to expand...

I am working on the above, but first of all I have to work out how I'm going to pay my rent this month.

Bill.
 
Another thing that Rupert T Gould goes into in his book Oddities, is what are called perfect numbers.

To recap a bit here: prime numbers are those integers (whole numbers) that are only divisible by themselves and one. All prime numbers are odd, with the exception of 2. Examples are 2,3,5,7,11,13 and so on. All other integers are composite numbers, that is they are divisible by at least one other integer. 4 is divisible by 2 and 1. 1 is always a factor. 8 by 4 and 2, and 1. 15 by 5 and 3 and 1. and so on and so forth.

What I have always found fascinating is 'perfect numbers', that is numbers that are the sum of their own factors. Perfect numbers are quite rare. There are just two below 100 and four below 1,000,000. They are 6, 28 496, and 8128.

All perfect numbers follow the formula 2p−1× (2p − 1) where p is prime. However not all numbers of this type are necessarily perfect numbers.

Take a look at 6. p in this case is 2. So, we have then 2 (2 to the 2nd power-1, or 2 to the first power) times (2 to the second power) or 4 minus 1 or 3. 2x3 obviously, is 6. When the sum of its' factors (2x3x1) are added up, we have six. So 6 is the equal to the sum of its' own factors. You can do the same thing where p=3. 2 to the 3rd minus one power (2 squared) times 2 to the third power minus 1 (8-1=7) or 4x7=28. This integer is also the sum of its' own factors. 14+7+4+2+1=28. I'd go on except that this has not been the easiest typing.

So, we have all perfect numbers following the formula 2p−1× (2p − 1), but not all numbers following this form are perfect numbers.

This, to me anyway, is quite fascinating. Hopefully it was of interest to at least some of you.

ETA: the 'p' in the formula above is supposed to be in superscript.
 
Gould also tells us of 'amicable numbers'. He gives the examples of 220, and 284. The sum of the factors of 220 is 284 and the sum of 284's factors is 220.

The same relationship exists between 1184 and 1210.

There may very well be others.
 
Reading Gould's book last night, I was struck by a curious fact. Could this be a set of amicable numbers not noticed previously? I refer to the definition of a prime number below

gerhard1 said:
To recap a bit here: prime numbers are those integers (whole numbers) that are only divisible by themselves and one. All prime numbers are odd, with the exception of 2. Examples are 2,3,5,7,11,13 and so on. All other integers are composite numbers, that is they are divisible by at least one other integer. 4 is divisible by 2 and 1. 1 is always a factor. 8 by 4 and 2, and 1. 15 by 5 and 3 and 1. and so on and so forth.
Click to expand...
Now, let's look at amicable numbers.
gerhard1 said:
Gould also tells us of 'amicable numbers'. He gives the examples of 220, and 284. The sum of the factors of 220 is 284 and the sum of 284's factors is 220.

The same relationship exists between 1184 and 1210.

There may very well be others
Click to expand...
Gould pointed out that the factors of 32 total 31. I checked this out and made certain that it is so. Not that I had any doubts about Gould, but I had to verify it. 31 is a prime number having only two factors; itself and one. When these are added we get 32. So would not 31 and 32 also be amicable numbers?

I checked the list of amicable numbers and the smallest ones named were 220 and 284.

Are primes precluded by definition from being amicable numbers?
 
gerhard1 said:
Reading Gould's book last night, I was struck by a curious fact. Could this be a set of amicable numbers not noticed previously? I refer to the definition of a prime number below


Now, let's look at amicable numbers.

Gould pointed out that the factors of 32 total 31. I checked this out and made certain that it is so. Not that I had any doubts about Gould, but I had to verify it. 31 is a prime number having only two factors; itself and one. When these are added we get 32. So would not 31 and 32 also be amicable numbers?

I checked the list of amicable numbers and the smallest ones named were 220 and 284.

Are primes precluded by definition from being amicable numbers?
Click to expand...
Did some more checking and this website at least does seem to preclude primes from consdderation.

Amicable numbers are two different numbers a and b so related that the sum of the proper divisors(*) of a is equal to the sum of the proper divisors of b and the sum of the proper divisors of b is equal to the sum of the proper divisors of a. (*) A proper divisor of a number is a positive integer divisor other than the number itself.
Click to expand...

http://www.vaxasoftware.com/doc_eduen/mat/numamigos_eng.pdf

Still it is curious.
 
gerhard1 said:
Did some more checking and this website at least does seem to preclude primes from consdderation.



http://www.vaxasoftware.com/doc_eduen/mat/numamigos_eng.pdf

Still it is curious.
Click to expand...

Hello Fora. Long time reader, first time poster.

It's the proper factors that are summed. You have to exclude the number itself or the sum will always be more than the number and you wouldn't get any amicable pairs or sociable chains. And the factors of perfect numbers would sum to twice the not-so-perfect number itself.
 
oxo66 said:
Hello Fora. Long time reader, first time poster.

It's the proper factors that are summed. You have to exclude the number itself or the sum will always be more than the number and you wouldn't get any amicable pairs or sociable chains. And the factors of perfect numbers would sum to twice the not-so-perfect number itself.
Click to expand...
Yes, of course; that makes sense. Thank you.
 
gerhard1 said:
Gould pointed out that the factors of 32 total 31. I checked this out and made certain that it is so. Not that I had any doubts about Gould, but I had to verify it. 31 is a prime number having only two factors; itself and one. When these are added we get 32. So would not 31 and 32 also be amicable numbers?
Click to expand...
Just as a sidenote, the sum of the factors of any power of 2 (8, 16, 32, 64 etc) will always be one less than the number itself.
 
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