mathreport - C.J. Gorrell Math 274 1 A Study on Number...

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C.J. Gorrell Math 274 1 A Study on Number Theory and Cryptography Donald Davis’ book The Nature and Power of Mathematics , is a book that covers key historical discoveries made in the world of mathematics, as well as a book that shows readers the possibilities that mathematics can present. Davis’ purpose for the book is to show readers the dazzling mathematics that schools so often neglect in their teaching of the subject. In sections three and four of the book, Davis covers the subjects of number theory, and cryptography. The parts of greatest interest of the number theory he discusses is on prime numbers, and congruent arithmetic. The reason these two focuses are of greater interest is because they are two principles that are of increasing importance in the later subject of study, cryptography. A prime number by definition is any number that is divisible by only the integers of one and itself, with the exception of the number 1. The study of prime numbers dates back to ancient Greece, and is still studied today. There is a very limited knowledge on prime numbers, because they display many irregular behaviors that we cannot explain. Finding primes has always been a centralized area of focus in math. While there is yet a system that could instantly or even very quickly determine if a number is prime or not, there are tedious, systematic ways to determine if a number is or is not prime. Davis explains in his book that in first determining if a number is prime or not it is important to note that all non-primes, composite numbers, are simply the product of primes. This is known as the Fundamental Theorem of Arithmetic. One of the first methods Davis presents to determine if a number is prime or not, is to simply try and divide each number less than the possible prime number, and see if the result leads to a combination of primes. While not to bad for small numbers, this becomes extremely tedious for larger numbers. The next method shown divides the workload in half. That is that if a number n is not divisible by any numbers less than the square root of n then n is a prime. While this is faster than the first method, it is still extremely slow for finding larger prime numbers. Eratosthenes formulated one of the most famous methods that existed for determining prime numbers in ancient Greece. This method of Eratosthenes is known as the Sieve of Eratosthenes. The sieve of Eratosthenes is a method that is extremely powerful in comparison to the brute force methods of determining primes. What the method involves it that in a list of consecutive integers from 2 on, starting with the lowest prime, each factor of that prime in the list is crossed out. Then moving to the next lowest prime the process is
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This note was uploaded on 01/24/2011 for the course MATH math274 taught by Professor Boyle during the Spring '10 term at Maryland.

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mathreport - C.J. Gorrell Math 274 1 A Study on Number...

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