In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.
> In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.
When written down as a string of digits, log n is another way to say 'proportional to the number of digits'.
The number of digits grows fairly slowly, thus also the 'probability' of a number being prime drops very slowly.
That’s what ‘everybody’ thinks. I think that’s from reading so much about them being hard to find. They aren’t hard to find, though, it’s (as far as we know) hard to recognize integers as being primes.
There are more prime numbers than there are squares of integers.
> I think that’s from reading so much about them being hard to find
More from the fact that each prime number makes all its multiples non-prime, so you'd expect this would accumulate quickly in making primes an increasingly rare find. Which is the case, but slower than intuition suggests.
They must both have the same cardinality, ℵ0, because they are both infinite subsets of the natural numbers, and so can each be laid out in order and paired with every natural number.
> all integers have a square, while not all integers are prime.
That’s true, but I don’t see how that’s an argument. “All integers have a prime ‘the nth prime’, while not all integers are squares” similarly is true, but not an argument as to which set is denser.
> They aren’t hard to find, though, it’s (as far as we know) hard to recognize integers as being primes.
Depends on what you mean by 'hard'. It's easy in the sense that we have algorithms to decide whether a number is prime or composite that take time polynomial in the space it takes to write down your number (ie polynomial in log n).
