Re: Primes in the interval (Problem)
- --- In firstname.lastname@example.org,
"djbroadhurst" <d.broadhurst@...> wrote:
> I have already explained why your error is unboundedOff-list, Sergey asked me for an example of a large error.
Let p[n] = 10^1000 + 453.
Then p[n-1] = 10^1000 - 1769 is the previous prime.
Sergey defines Q as the number of primes
between p[n-1]^2 and p[n]^2.
Using the prime number theorem, we may estimate
Q =~ (p[n]^2 - p[n-1]^2)/log(p[n]^2) =~ 9.65*10^999
Sergey defines his "error" E by
E = (p[n]^2 - p[n-1]^2)*prod(k=1,n,1-1/p[k]) - Q
Using Mertens' theorem
we may easily estimate
E =~ 2*exp(-Euler)*Q - Q =~ 1.19*10^999
whose integer part has 1000 decimal digits.
Clearly the error is unbounded, since it is
of the same order as the n-th prime, p[n].
I remark that Sergei has already advertised his "blogspot" in 9
messages to this list, while making no significant contribution
to our own discussions. I hope that we shall not receive a 10th
advertisement for a site that seems to make no attempt to learn
from careful correction of its obvious failures.