1205 words - 5 pages

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

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wrong about the numbers 23, 29, 37, but was correct for the number 31. Number theorists have used many methods in order to find prime numbers; 2p-1 is not the only method that can be used (Caldwell).

To find small prime numbers from 1 to 100, there is a method that can be used called the Sieve of Eratosthenes. First, start out with a table of numbers from 1-100 shown in the table below:

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2

3

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5

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100

Since 2 is the smallest prime number, the objective is to get rid of all composites that are divisible by 2. The arrangement then becomes:

1

2

3

5

7

9

11

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23

25

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29

31

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The next smallest prime is 3, so now the objective is to get rid of all composites that are divisible by 3. Once you’re done with 3, you go on with 5, then 7. Since the square root of 100...

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