- Mathematicians studying prime numbers have remarked on the duality of their behavior.

For example, Andrew Odlyzko in his 2006 IRMACS lecture, states that "even though the

primes are very deterministic, we are conjecturing that they behave like random objects . .

. determinism on one side and random behavior on the other."

And Don Zagier, in an oft-quoted statement, has this to say:

"There are two facts about the distribution of prime numbers of

which I hope to convince you so overwhelmingly that they will be

permanently engraved in your hearts.

The first is that, despite their simple definition and role as the building blocks of the

natural numbers, the prime numbers belong to the most arbitrary and ornery objects

studied by mathematicians: they grow like weeds among the natural numbers, seeming to

obey no other law than that of chance, and nobody can predict where the next one will

sprout.

The second fact is even more astonishing, for it states just the

opposite: that the prime numbers exhibit stunning regularity, that there are laws

governing their behaviour, and that they obey these laws with the almost military

precision."

My question is: Why is this duality more surprising than that found in probability theory,

wherein the distribution of random events is regular (as the normal curve) while the actual

sequence remains random and unpredictable.

My understanding is that the PNT provides a good estimate of the distribution of primes

but that it is of no help in determining the nth prime (just as the bell-shaped curve does

not help us predict the next toss of a coin).

So -- anybody wanna take a crack at clearing up the confusion of a non-mathematician?

Thanks - 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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