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Why are primes so common? I have trouble rationalizing that even when you get into the range of millions and billions, you still see primes appear just a few hundred numbers apart. I would think the probability of primes appearing once you get to very large numbers would be extremely low, considering the amount of possible whole number divisors for a whole number n is equal to n - 2, not counting 1 and the number itself. It seems odd to me that primes appear so closely even in the billions. I mean, there are billions of possible numbers to divide from and you're telling me none of them divide evenly at all? It would make sense for this event to occur once every say million numbers, but every hundred just seems very close to me.

I don't study mathematics. This thought has just been lingering in my head for a while, and I would like to hear someone who knows what they are talking about explain to me why primes appear so closely. And please explain it in a way that makes sense to someone who isn't familiar with higher level mathematical terminology, or explain advanced terminology if you must use it.

Thank you!

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  • $\begingroup$ The smallest prime divisor of a number on the order of one billion has to be no larger than the order of 100,000. There aren't that many primes below 100,000 to go around for those billions of numbers. $$$$ And the asymptotic density of the primes is small once you're considering large enough numbers, by the prime number theorem. $\endgroup$ – user296602 Oct 24 '17 at 16:01
  • $\begingroup$ Ultimately the answer to this is the Prime Number Theorem, but that is quite a deep result. Chebyshev did prove a weak form of PNT, stating that the number of primes up to $x$ is at least a constant times $x/\ln x$. This is not nearly such a deep result, but still requires considerable ingenuity. $\endgroup$ – Lord Shark the Unknown Oct 24 '17 at 16:01
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Just to answer your question on why we find a prime number after every few hundred numbers in a billion: A billion is just not a big enough number itself. I am sure there will be mathematical proofs that relate to the frequency of prime numbers. However, consider the following facts.

  • There are more prime numbers than found in any list of primes.
  • Every natural number is divisible by at least one unique set of prime numbers.
  • There are infinitely many of both natural numbers and prime numbers.

Now think about this: if the only numbers we are looking at are from 1-7; we might find it weird that most numbers are prime. Or if we look at just the numbers from 1-100, we could say that 25% of all numbers are prime numbers and they are not so special.

We have primes like (3,5), (11,13), (17,19)... We think that they are special because the difference is just 2 and call them the twin primes. Of all I know, maybe there is a set of a billion prime numbers with a difference of exactly 1 billion.

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    $\begingroup$ The first bullet point you made really made it make more sense for me. I guess I kind of forgot that primes are sort of like atoms and make up all other numbers. So must be a certain undefined ratio of them to natural numbers for this proven law to hold true. $\endgroup$ – James G Oct 24 '17 at 17:39
  • $\begingroup$ Interesting analogy. Prime numbers are like elements and non-primes are like compounds made from other primes. $\endgroup$ – FUBAR Oct 25 '17 at 18:13

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