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?
Best Answer
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.