Another question from a non-mathematician

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• Mathematicians studying prime numbers have remarked on the duality of their behavior. For example, Andrew Odlyzko in his 2006 IRMACS lecture, states that even
Message 1 of 3 , Jun 1, 2007
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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

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
Message 2 of 3 , Jun 2, 2007
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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
Message 3 of 3 , Jun 4, 2007
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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