- I think the randomness and bell curve are in one sense much more

predictable than primes:

if you flip a coin long enough, you are eventually going to get N

heads in a row for sure.

But with primes, we don't know whether or not in the long run you will

get infinitely many

twin primes or not.

The problem is, really random things will actually do everything in

the long run. But the primes aren't random!

That is, the bell curve describes what random variables tend to do in

the long run (namely, be off the mean by about sqrt(n)).

But while the prime number theorem is similar for primes, since the

primes ARE predictable we need to understand a lot more about how they

deviate in the short run from their long run average.

--Joshua ZUcker - Mathematics is linked with determinism and

predictability. Thus the seeming randomness of primes

is striking. The duality is what makes the primes so

mysterious and interesting. If you google 'duality

prime numbers', you will find this paper

(www.secamlocal.ex.ac.uk/

people/staff/mrwatkin/isoc/huang.pdf)

that provides an elementary proof for the duality of

primes.

--- Joshua Zucker <joshua.zucker@...> wrote:

> I think the randomness and bell curve are in one

____________________________________________________________________________________

> sense much more

> predictable than primes:

> if you flip a coin long enough, you are eventually

> going to get N

> heads in a row for sure.

>

> But with primes, we don't know whether or not in the

> long run you will

> get infinitely many

> twin primes or not.

>

> The problem is, really random things will actually

> do everything in

> the long run. But the primes aren't random!

>

> That is, the bell curve describes what random

> variables tend to do in

> the long run (namely, be off the mean by about

> sqrt(n)).

>

> But while the prime number theorem is similar for

> primes, since the

> primes ARE predictable we need to understand a lot

> more about how they

> deviate in the short run from their long run

> average.

>

> --Joshua ZUcker

>

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