How Many Primes Are Known?

Date: 01/21/2003 at 19:05:29
From: Jones Thomas
Subject: Primes

I know that there is an infinite number of primes, but how many are 
currently known?

Date: 01/22/2003 at 10:29:06
From: Doctor Kastner
Subject: Re: Primes

Hi Jones -

I don't know of any official count for the number of primes currently 
known. Part of the reason may be that they are too numerous; given a 
reasonably sized interval of integers, there will almost certainly be 
a prime or two lurking in there. Actually, one of the open questions 
about prime numbers is whether there is always a prime between n^2 and 
(n+1)^2.

We do have a useful tool in the Prime Number Theorem, which says that 
given a number n, the number of primes less than or equal to it is 
about

   n
-------
 ln(n)

There have been lots of refinements to this fraction that give a 
better approximation, but this is an easy one to remember. 

Instead of asking "how many primes are there," people have instead 
asked, "how many primes of a special form are there?" For instance, 
consider the function

f(n) = n^2 +1

which produces primes for n = 2,4,6,10... it is not known whether 
there is an infinite number of primes of this form. Similarly, 
consider the function

f(n) = 2^n -1

which produces primes for n = 2,3,5,7,13,17,19,31... These primes are 
called Mersenne primes and it is thought that an infinite number of 
them exist. Right now however, the list of Mersenne primes contains 
only 39 entries, and that's a far cry from infinity!

I hope this helps.  Write back if you have more questions.

- Doctor Kastner, The Math Forum
  http://mathforum.org/dr.math/