Are there more integers then prime numbers? I kind of feel this question may have been asked in some way before, but I could not find it.
I know there are infinite prime numbers (because Euclid tells us), and there are infinite integers. For any given range of posative whole numbers there are always more integers then there are prime numbers.
So it appears that there should be "more" integers in total then there are prime numbers. Is this statement true? How do you compare two infinite series when one is a subset of the other?
 A: The sets have the same cardinality, so there are "as many" integers as primes.
However, you can consider the density of the primes as a subset of the integers.  (This is different from considering the set of prime numbers itself, which is indistinguishable from any other countable set.)  This characterization is given by the prime number theorem, which you can read about on Wikipedia.
A: Prime numbers and integers have the same cardinality which means there is a bijection between integers and prime numbers.
In other words: For every prime number you are able to find an related integer and vice versa.
A: Well there are more than one ways of infinity. In this case both integers and prime numbers are "countable" - meaning there are as many of integers as prime numbers.
You can take a look here: 
http://en.wikipedia.org/wiki/Countable_set
To be a bit clearer: 
A set is countable, if can find a surjective transformation from the natural numbers to that set. 
This is the case for both integers and primes are therefore both integers and primes are countable (and have the same "number of elements" or better "type of infinite number of elements" = cardinality)
A: By the measure of cardinality, which it is most conventional to use when measuring infinite sets, there are the same number of prime numbers as there are integers.  This measure is defined by being able to place the sets in bijection which means you can place them into 1:1 correspondence and never have either a prime left over with no corresponding natural number, nor can you have a natural number left over with no corresponding prime.
However there are those who would argue that identity is a stronger measure of correspondence than bijection.  And by this argument we can see that the prime numbers are a proper subset of the integers and therefore no matter how many prime numbers we might count, every single one will not just be in $1:1$ correspondence with an integer, it it will actually be an integer, and will be in the set of integers.  The set of integers contains all of those and all of the compound numbers on top.
This second measure is best captured by the notion of density.  We can say that the integers have density $1$ in themselves while the prime numbers have a density lower than $1$ among the integers.
Cardinality is useful when we need to compare sizes of infinite sets and we're unable to compare them by the identities of the elements within, which is frequently the case.
If trees continue producing apples and oranges for eternity, density will correctly measure the fact that the fruit basket would need to be bigger if you planned to pick both types of fruits.  Cardinality on the other hand would simply tell you that you need a really big basket either way.
