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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) ha

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.

eru 1y ago
> 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).
throw310822 1y ago
> 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.
knome 1y ago
>There are more prime numbers than there are squares of integers. all integers have a square, while not all integers are prime. in any given span, you'll see more primes than squares, however. more dense?
- madcaptenor 1y ago
  For example, the number of primes less than n is around n/log(n) while the number of squares less than n is around sqrt(n), which is much smaller.
- SamBam 1y ago
  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.
- Someone 1y ago
  > 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.

eru 1y ago

> 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).

throw310822 1y ago

> 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.

knome 1y ago

>There are more prime numbers than there are squares of integers. all integers have a square, while not all integers are prime. in any given span, you'll see more primes than squares, however. more dense?

madcaptenor 1y ago
For example, the number of primes less than n is around n/log(n) while the number of squares less than n is around sqrt(n), which is much smaller.
SamBam 1y ago
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.
Someone 1y ago
> 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.