Changes

The Prime Number Theorem is one of the most famous theorem in mathematics.  It states that the number of primes not exceeding n is asymptotic to n/log(n), where log(n) is the logarithm of (n) to the base e.   

The Prime Number Theorem is one of the most famous theorem in mathematics.  It states that the number of primes not exceeding n is asymptotic to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e.   

The number of primes not exceeding n is commonly written as pi(n), and an asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n).  (This does not mean that a(n)-b(n) is small as n increases.  It means the ratio of a(n) to b(n) approaches one as n increases.)

The number of primes not exceeding n is commonly written as <math>\pi(n)</math>, and an asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n).  (This does not mean that a(n)-b(n) is small as n increases.  It means the ratio of a(n) to b(n) approaches one as n increases.)

The Prime Number Theorem thus states that pi(n) ~ n/log(n) .

The Prime Number Theorem thus states that <math> \pi(n) </math>~<math> n/ \log(n)</math> .

In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one.  Put a third way, n/log(n) is a good approximation for pi(n).

In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one.  Put a third way, n/log(n) is a good approximation for <math>\pi(n)</math>.

 

 

 

 

In fact, for large x this turns out to be a better approximation than <math>\pi(x)</math>.