Man, That Is One Prime Theorem!
Abstract
As much of the mathematical community knows, infinitely many prime numbers exist. An elementary number theory
proof demonstrates that arbitrarily long sequences of consecutive natural numbers in which all numbers are
composite exist. On the other end lies the popular conjecture that infinitely many “twin primes,” prime
numbers separated by 2, such as 3 and 5, 17 and 19, or 101 and 103, exist. This paper proves the Twin Prime
Conjecture. Well, actually it traces the proof of the Prime Number Theorem, which states that the number of
primes not exceeding
x is asymptotic to
x/log x.
That is, if we define
(x) =
|{
p 
: p≤x, p prime}
|,
then the density of primes
(x)/x
asymptotically approaches
1/log x. With a theorem asserting the thinning of
prime numbers and a conjecture asserting they occasionally lie close together infinitely often, the Prime
Number Theorem offers insight into just how much the distribution of primes thin as we count arbitrarily high.
Table of Contents
Complete List of References
- [HG1]
Hardy, G.H. and Wright, E.M.
An Introduction to the Theory of Numbers.
Oxford: Clarendon Press, 1960.
- [WE1]
Weisstein, Eric.
Prime Number Theorem.
From MathWorld—A Wolfram Web Resource.
