How can we find a large prime number

People use numbers whenever they do math. Yet, do they know that each number in the number system has its own unique trait? Numbers such as 4 and 9 are considered square numbers because 2 times 2 is 4, and 3 times 3 is 9. There also prime numbers. Prime numbers are numbers that have exactly two divisors. The number one is not included because it only has one divisor, itself. The smallest prime number is two, then three, then five, and so on. This list goes on forever and the largest known primes are called Mersenne primes. A Mersenne prime is written in the form of 2p-1. So far, the largest known Mersenne prime is 225,964,951-1, which is the 42nd Mersenne prime. This prime number has 7,816,230 digits!

Many number theorists, who study certain properties of integers, have been trying to find formulas to generate primes. They believed that 2p-1 would always generate primes whenever p is prime. It turns out that if p is composite, then the number will also be a composite number. However, later mathematicians claimed that 2p-1 only works for certain primes p. For example, the number 11 is a prime because its divisors are only 1 and 11. In this case, 211-1 is 2047 and Hudalricus Regius showed that this number is composite in 1536 because 23 and 89 are factors of 2047. From then on, whenever a prime number can be written in the form of 2p-1, it is considered to be a Mersenne prime. Many conjectures have been made about p. Pietro Cataldi showed that 2p-1 was true for 17 and 19. However, he stated that it was also true for the prime numbers 23, 29, 31, and 37. Number theorists such as Fermat and Euler proved that Cataldi was

2

wrong about the numbers 23, 29, 37, but was correct for the number 31....

... middle of paper ...

...a that can generate primes. Since there is an infinite amount of primes, we cannot conclude what the largest prime is. However, we do know that there are 25 primes less than 100, 168 less than 1000, 1229 less than 10,000, and as of January 2000, there

5

are 2,220,819,602,560,918,840 primes less than 100,000,000,000,000,000,000 (Flannery 69). However, by working with other people, perhaps we can use all of these methods to discover the next largest prime.

6

Works Cited

Caldwell, Chris K. Mersenne Primes: History, Theorems and Lists. 29 July 2005.

Flannery, Sarah. In Code. Chapel Hill: Algonquin Books of Chapel Hill, 2001.

Mersenne Prime Search. 06 March 2005. GIMPS. 27 July 2005.

Robert, A. Wayne and Dale E. Varberg. Faces of Mathematics. New York: Harper & Row Publishers, Inc., 1978.

People use numbers whenever they do math. Yet, do they know that each number in the number system has its own unique trait? Numbers such as 4 and 9 are considered square numbers because 2 times 2 is 4, and 3 times 3 is 9. There also prime numbers. Prime numbers are numbers that have exactly two divisors. The number one is not included because it only has one divisor, itself. The smallest prime number is two, then three, then five, and so on. This list goes on forever and the largest known primes are called Mersenne primes. A Mersenne prime is written in the form of 2p-1. So far, the largest known Mersenne prime is 225,964,951-1, which is the 42nd Mersenne prime. This prime number has 7,816,230 digits!

Many number theorists, who study certain properties of integers, have been trying to find formulas to generate primes. They believed that 2p-1 would always generate primes whenever p is prime. It turns out that if p is composite, then the number will also be a composite number. However, later mathematicians claimed that 2p-1 only works for certain primes p. For example, the number 11 is a prime because its divisors are only 1 and 11. In this case, 211-1 is 2047 and Hudalricus Regius showed that this number is composite in 1536 because 23 and 89 are factors of 2047. From then on, whenever a prime number can be written in the form of 2p-1, it is considered to be a Mersenne prime. Many conjectures have been made about p. Pietro Cataldi showed that 2p-1 was true for 17 and 19. However, he stated that it was also true for the prime numbers 23, 29, 31, and 37. Number theorists such as Fermat and Euler proved that Cataldi was

2

wrong about the numbers 23, 29, 37, but was correct for the number 31....

... middle of paper ...

...a that can generate primes. Since there is an infinite amount of primes, we cannot conclude what the largest prime is. However, we do know that there are 25 primes less than 100, 168 less than 1000, 1229 less than 10,000, and as of January 2000, there

5

are 2,220,819,602,560,918,840 primes less than 100,000,000,000,000,000,000 (Flannery 69). However, by working with other people, perhaps we can use all of these methods to discover the next largest prime.

6

Works Cited

Caldwell, Chris K. Mersenne Primes: History, Theorems and Lists. 29 July 2005.

Flannery, Sarah. In Code. Chapel Hill: Algonquin Books of Chapel Hill, 2001.

Mersenne Prime Search. 06 March 2005. GIMPS. 27 July 2005.

Robert, A. Wayne and Dale E. Varberg. Faces of Mathematics. New York: Harper & Row Publishers, Inc., 1978.

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