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Prime Number Theorem (view source)
Revision as of 02:03, December 25, 2006
, 02:03, December 25, 2006no edit summary
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 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 Prime Number Theorem thus states that pi(n) ~ n/log(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 pi(n).
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 Prime Number Theorem thus states that pi(n) ~ n/log(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 pi(n).