| ▲ | throw310822 3 days ago |
| Kind of surprising, my intuitive idea of primes is that they become rarer much faster, while there's really a ton of them. |
|
| ▲ | susam 3 days ago | parent | next [-] |
| They indeed do become rarer. Plotting all the primes in a single row makes this apparent, like so: https://susam.net/primegrid.html#1-1-1000000 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. I have a small section about this at https://susam.net/journey-to-prime-number-theorem.html#prime... if you want to read more about this. See also: https://en.wikipedia.org/wiki/Prime_number_theorem |
| |
| ▲ | eru 3 days ago | parent [-] | | Yes. > 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. |
|
|
| ▲ | kristopolous 3 days ago | parent | prev | next [-] |
| This is a very well researched topic https://en.m.wikipedia.org/wiki/Prime_number_theorem |
|
| ▲ | Someone 3 days ago | parent | prev | next [-] |
| 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. |
| |
| ▲ | throw310822 3 days ago | parent | next [-] | | > 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 3 days ago | parent | prev | next [-] | | >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? | | |
| ▲ | Someone a day ago | parent | next [-] | | > 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. | |
| ▲ | SamBam 3 days ago | parent | prev | next [-] | | 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. | |
| ▲ | madcaptenor 3 days ago | parent | prev [-] | | 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. |
| |
| ▲ | eru 3 days ago | parent | prev [-] | | > 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). |
|
|
| ▲ | leric 3 days ago | parent | prev [-] |
| [dead] |
