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### 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

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.

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/
```
Associated Topics:
High School Number Theory
Middle School Prime Numbers

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