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 nonprimes, 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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 Spring '10
 boyle
 Prime Numbers, Prime number, Publickey cryptography, Donald Davis, C.J. Gorrell Math

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