Saturday, October 6, 2007
Something About Prime Numbers
Rekindling my liking for Maths, I thought of taking a tour of prime numbers! I wanted to find out the largest mersenne prime number. I ended up browsing through many prime number sites and gathered a bit of information.
The beauty of prime numbers is elegantly shown by the fundamental theorem of arithmetic: every positive integer greater than one can be expressed uniquely as a product of primes, apart from the rearrangement of terms (Examples: 15 = 3 * 5, 150 = 2 * 3 * 5 * 5). This theorem was formulated by Euclid way back in about 350 BC. When thinking about axioms in geometry (learned way back in secondary school) it is this person that comes into mind. In fact, he is often called as the father of geometry.
There are infinite number of primes. It can be proved in a few lines as follows:
Assume n is the highest known prime number.
Lets define m = (2 * 3 * 5 * 7 * 11 * 19 *...*n) + 1 (i.e. multiplication of all the prime numbers up to n plus 1)
Now, if you divide m by any known prime number, the reminder will always be 1.
So m is a prime number.
Obviously m > n.
So this contradicts our assumption.
Conclusion: there are infinite number of primes.
There is an interesting subset of primes called "Mersenne Primes" (named after the French philosopher Marin Mersenne). Mersenne primes have this special property that the value is equal to one less than a prime power of two. In mathematical terms it is equal to (2^n - 1) where n is a prime number. The smallest mersenne prime is 3 (2^2 - 1). Successive ones are 7 (2^3 - 1), 31 (2^5 - 1), 127 (2^8 -1) and the list continues.
Until the 20th century, only a handful of mersenne primes were found. Most of the known mersenne primes were discovered after 1950. Mathematicians have so far found 43 mersenne primes altogether. The 43rd one was discovered in December last year. This was discovered through the GIMPS by Dr. Curtis Cooper and Dr. Steven Boone, professors at Central Missouri State University. The number is (2^30402457-1). It has 9152052 digits! The GIMPS, Great Internet Mersenne Prime Search, is an on-going distributed computing project on the Internet. In fact, the last 9 mersenne primes were all discovered by the mathematicians at GIMPS.
Although there are infinite number of primes, it is not known whether there is an upper limit for mersenne primes. The game is still active searching tfor he next mersenne primes. In addition to making history, you can earn a lump sum! The Electronic Frontier Foundation is offering a $100,000 award (really huge amount in Indian rupees) to the first person who discovers a ten million digit prime number.
This reminds me about one of the quirky tests of Google. They put up a billboard on Route 101 that read, in its entirety "{first 10-digit prime found in consecutive digits of e}.com". No Google logo, no recruiting pitch. Just the equation. The curious who solved it (the answer is 7427466391.com) and went to that web page and started their first step towards becoming a Google employee! Math can really pay you off!!
The beauty of prime numbers is elegantly shown by the fundamental theorem of arithmetic: every positive integer greater than one can be expressed uniquely as a product of primes, apart from the rearrangement of terms (Examples: 15 = 3 * 5, 150 = 2 * 3 * 5 * 5). This theorem was formulated by Euclid way back in about 350 BC. When thinking about axioms in geometry (learned way back in secondary school) it is this person that comes into mind. In fact, he is often called as the father of geometry.
There are infinite number of primes. It can be proved in a few lines as follows:
Assume n is the highest known prime number.
Lets define m = (2 * 3 * 5 * 7 * 11 * 19 *...*n) + 1 (i.e. multiplication of all the prime numbers up to n plus 1)
Now, if you divide m by any known prime number, the reminder will always be 1.
So m is a prime number.
Obviously m > n.
So this contradicts our assumption.
Conclusion: there are infinite number of primes.
There is an interesting subset of primes called "Mersenne Primes" (named after the French philosopher Marin Mersenne). Mersenne primes have this special property that the value is equal to one less than a prime power of two. In mathematical terms it is equal to (2^n - 1) where n is a prime number. The smallest mersenne prime is 3 (2^2 - 1). Successive ones are 7 (2^3 - 1), 31 (2^5 - 1), 127 (2^8 -1) and the list continues.
Until the 20th century, only a handful of mersenne primes were found. Most of the known mersenne primes were discovered after 1950. Mathematicians have so far found 43 mersenne primes altogether. The 43rd one was discovered in December last year. This was discovered through the GIMPS by Dr. Curtis Cooper and Dr. Steven Boone, professors at Central Missouri State University. The number is (2^30402457-1). It has 9152052 digits! The GIMPS, Great Internet Mersenne Prime Search, is an on-going distributed computing project on the Internet. In fact, the last 9 mersenne primes were all discovered by the mathematicians at GIMPS.
Although there are infinite number of primes, it is not known whether there is an upper limit for mersenne primes. The game is still active searching tfor he next mersenne primes. In addition to making history, you can earn a lump sum! The Electronic Frontier Foundation is offering a $100,000 award (really huge amount in Indian rupees) to the first person who discovers a ten million digit prime number.
This reminds me about one of the quirky tests of Google. They put up a billboard on Route 101 that read, in its entirety "{first 10-digit prime found in consecutive digits of e}.com". No Google logo, no recruiting pitch. Just the equation. The curious who solved it (the answer is 7427466391.com) and went to that web page and started their first step towards becoming a Google employee! Math can really pay you off!!
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