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(Math) Prime Numbers

 
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adedios
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Joined: 06 Jul 2005
Posts: 5060
Location: Angel C. de Dios

PostPosted: Wed Jan 04, 2006 1:32 pm    Post subject: (Math) Prime Numbers Reply with quote






Mo. Researchers Find Largest Prime Number
By GARANCE BURKE
Associated Press Writer
Tue Jan 3, 10:09 PM ET

Researchers at a Missouri university have identified the largest known prime number, officials said Tuesday.

The team at Central Missouri State University, led by associate dean Steven Boone and mathematics professor Curtis Cooper, found it in mid-December after programming 700 computers years ago.

A prime number is a positive number divisible by only itself and 1 — 2, 3, 5, 7 and so on.

The number that the team found is 9.1 million digits long. It is a Mersenne prime known as M30402457 — that's 2 to the 30,402,457th power minus 1.

Mersenne primes are a special category expressed as 2 to the "p" power minus 1, in which "p" also is a prime number.

"We're super excited," said Boone, a chemistry professor. "We've been looking for such a number for a long time."

The discovery is affiliated with the Great Internet Mersenne Prime Search, a global contest using volunteers who run software that searches for the largest Mersenne prime.

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Related Lessons (Elementary Level)


http://www.sciencenetlinks.com.....;DocID=138

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Questions to explore further this topic:

Are there different types of numbers?

http://www.theeducationwebsite.....umbers.pdf
http://www.math.utah.edu/online/1010/numbers/
http://academic.cuesta.edu/sma.....2/act2.htm

Are there different types of whole numbers?

http://www2.ups.edu/community/.....actors.htm

What is the difference between prime and composite numbers?

http://www.mathgoodies.com/les.....ility.html

All about whole numbers (and integers):

http://www.1729.com/applets/izone.html

What are prime numbers?

http://mathforum.org/dr.math/f.....e.num.html
http://www.factmonster.com/ipka/A0876084.html
http://www.icteachers.co.uk/ch.....primes.htm
http://primes.utm.edu/primes/b.....uction.php

What is prime factorization?

http://www.brainpop.com/math/p.....orization/
http://www.thirteen.org/edonli.....ctors.html


What are the largest known prime numbers?

http://primes.utm.edu/largest.html#biggest

What are multiples?

http://www.mathgoodies.com/lessons/vol3/lcm.html

What are powers of two?

http://www.mathgoodies.com/les.....nents.html

Lessons and Activities on numbers:

http://www.bbc.co.uk/schools/r.....hs/number/

Here is an interesting note regarding prime numbers and a biological cycle:

http://www.abc.net.au/science/.....421251.htm

GAMES

http://www.kidsdomain.com/games/math2.html
http://illuminations.nctm.org/.....aspx?id=12
http://illuminations.nctm.org/.....aspx?id=29
http://www.quia.com/cb/8436.html


Last edited by adedios on Sat Jan 27, 2007 3:06 pm; edited 2 times in total
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saida



Joined: 07 Jul 2005
Posts: 840
Location: Saida Cagandahan Dulay

PostPosted: Wed Jan 04, 2006 11:54 pm    Post subject: Re: Science Lesson: Mo. Researchers Find Largest Prime Numbe Reply with quote

adedios wrote:

The number that the team found is 9.1 million digits long. It is a Mersenne prime known as M30402457 — that's 2 to the 30,402,457th power minus 1.


WOW, i'm so curious kung ano yung 9.1 million digits na yun! samantalang noon ay hirap na hirap akong tandaan man lang kung alin yung composite at alin yung prime...<lol>
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Saida Cagandahan Dulay
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tony.basa
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PostPosted: Thu Jan 05, 2006 6:50 am    Post subject: Reply with quote

Quote:

The discovery is affiliated with the Great Internet Mersenne Prime Search, a global contest using volunteers who run software that searches for the largest Mersenne prime.


At one time, my PC was participating but I found the program intrusive. Rather than letting me do my stuffs as if nothing is happening in the backgroud, I found my apps not responsive.

IIRC, it's being hosted by the same site where SETI has moved since.

cheers!
tony.basa
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adedios
SuperPoster


Joined: 06 Jul 2005
Posts: 5060
Location: Angel C. de Dios

PostPosted: Mon May 21, 2007 8:37 pm    Post subject: A mighty number falls Reply with quote

Ecole Polytechnique Fédérale de Lausanne

A mighty number falls
21 May 2007

Mathematicians and number buffs have their records. And today, an international team has broken a long-standing one in an impressive feat of calculation.

On March 6, computer clusters from three institutions – the EPFL, the University of Bonn and NTT in Japan -- reached the end of eleven months of strenuous calculation, churning out the prime factors of a well-known, hard-to-factor number that is a whopping 307 digits long.

"This is the largest 'special' hard-to-factor number factored to date," explains EPFL cryptology professor Arjen Lenstra. (The number is 'special' because it has a special mathematical form -- it is close to a power of two.) The news of this feat will grab the attention of information security experts and may eventually lead to changes in encryption techniques.

Although it is relatively easy to identify huge prime numbers, factoring, or breaking a number down into its prime components, is extremely difficult. RSA encryption, named for the three individuals who devised the technique (Ronald Rivest, Adi Shamir and Leonard Adleman), takes advantage of this. Using the RSA method, information is encrypted using a large composite number, usually 1024 bits in size, created by multiplying together two 150-or-so digit prime numbers. Only someone who knows those two numbers, the "keys", can read the message. Because there is a vast supply of large prime numbers, it's easy to come up with unique keys. Information encrypted this way is secure, because no one has ever been able to factor these huge numbers. At least not yet.

The most recent factoring record is RSA200, a 200-digit 'non-special' number whose two prime factors were identified in 2005 after 18 months of calculations that took over a half century of computer time.

The international team factored the current 307-digit behemoth using the "special number field sieve," a method devised in the late 1980s by Lenstra (then at Bellcore), his brother Hendrik, then a professor at UC Berkeley, English mathematician John Pollard and Mark Manasse from DEC. The 11-month job took a century of computer time.

A feat like this would have been unthinkable back in 1990 when Lenstra started applying number theory and distributed computing to the task of breaking factoring records. Increased computer power and refined computational techniques have raised the bar, and will continue to do so. "We have more powerful computers, we have come up with better ways to map the algorithm onto the architecture, and we take better advantage of cache behavior," Lenstra explains.

Is the writing on the wall for 1024-bit encryption" "The answer to that question is an unqualified yes," says Lenstra. For the moment the standard is still secure, because it is much more difficult to factor a number made up of two huge prime numbers, such as an RSA number, than it is to factor a number like this one that has a special mathematical form. But the clock is definitely ticking. "Last time, it took nine years for us to generalize from a special to a non-special hard-to factor number (155 digits). I won't make predictions, but let's just say it might be a good idea to stay tuned."
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