Re: sketch of a proof of twin prime conjecture....it can be done
- From: Dan Cass <dcass@xxxxxxxx>
- Date: Fri, 04 Dec 2009 07:54:49 EST
On Dec 3, 2:32 pm, Dan Cass <dc...@xxxxxxxx> wrote:
wrote:On Dec 2, 10:42 am, Dan Cass <dc...@xxxxxxxx>
that n<stathopoulo...@xxxxxxxxx>On 2 Dec, 14:34, eestath
It might be easier to figure the probability
is
NOT divisible by a prime.
Only over all primes less than
If one takes your criticism:> Not over ALL primes.
zero.sqrt(n)
seriously, then the product in question is not
But that is a ridiculous thing to say, since youhave
changed my definition when you restrict to primesless than n.
The product I refer to is indeed over ALL primes.place.
Otherwise it wouldn't diverge to 0 in the first
The product over primes less than sqrt(n) still goes
to 0
in the limit as n-->oo. It is this limit to which I
referred.
The original question said "It might be easier to
figure the
probability that n
is NOT divisible by a prime."
n is an integer. therefore it is finite. therefore
the "probability"
that it is not divisible by a prime is
(1-1/2)(1-1/3)(1-1/5) ....(1-1/
p)
for p <= sqrt(n). But this is wrong.
The difficulty with the product argument (and the
reason that there is
no
sieve method proof of PNT is the same reason why the
constant in
Mertens' thm is exp(gamma) and not 1. The rough
probability that n is
prime
by the PNT is 1/log(n), whereas Mertens' Thm gives
exp(gamma)/log(n)
for the above product.
Once 2*3*5*7* ...... > n, i.e. once you have
enough terms
in the product (1-1/2)(1-1/3)... . so that the
product of the
primes EXCEEDS n, the probability that the NEXT
prime (say) q
does not divide n is no longer INDEPENDENT of the
probability that
the primes in the product divide q.
All I said in my response clearing things up was that,
if the statement "n has property P with probability s"
is interpreted as saying that the limit as N goes to
infinity of the proportion of n in the set {1,2,...,N}
which have the property P is s.
the 'n' here is not a specific integer.
It is to be viewed as a random variable which ranges
over the set {1,2,...,N}, and then if some probability
statement about the random variable n is such that,
as N --> oo, the probability has a limit, then
one can say "the probability of n having property P is s"
and this means only that the limiting probability as N
approaches infinity of the probability is s.
.
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