Re: Q: About number of primes with n digits?


Chip Eastham wrote:

Danny wrote:
Chip Eastham wrote:

Danny wrote:
jankrihau@xxxxxxxxxxx wrote:

Danny wrote:
The first 4 primes are single digits in length.
The next 21 primes are 2 digits in length.
The next 143 are 3 digits in length.
etc..

4, 21, 143, 1061, 8363, 68906, 586081, 5096876, 45086079,
404204977, 3663002302, 33489857205, 308457624821, 2858876213963,
26639628671867, 249393770611256, 2344318816620308,
22116397130086627, 209317712988603747, 1986761935284574233,
18906449883457813088, 180340017203297174362

Sequence is in OEIS as A006879.

Will the ratio between terms converge?

If the sequence is divergent then at any point can the next
ratio be < the previous ratio?

Dan

By the PNT, the nth term is asymptotically

0.9 * 10^n / (n log 10)

so the ratio converges to 10.

---
J K Haugland
http://home.no.net/zamunda

My reasoning is probably way off but if the
above formula you give is only asympototically
correct how can the ratio be an absolute convergence
too 10?

I have been looking up the PNT and see nothing
about any value pertaining to the different (n) lengths
of the primes. Only the different methods used for estimating
pi(x) not digit length (n) counting.
Also nothing about ratio convergence of this particular
count.

Then again, I could have over looked something.

Thanks,

Dan

The prime counting function pi(x) is defined to be the
number of (positive) primes less than or equal to x,
for any real number x. The Prime Number Theorem
says simply that pi(x) ~ x/ln(x), but a more precise
statement can be given in terms of the logarithmic
integral, esp. if one assumes the Riemann Hypothesis.

If the asymptotic character of this result bothers you,
a simple restatement is that pi(x)/(x/ln(x)) tends to 1
as x tends to +infinity.

In any case, isn't it evident that the "primes of length
k digits" is exactly:

pi(10^k) - pi(10^(k-1))

for integer k > 1?

regards, chip

Yes, it is evident!

In your first statement, even if assuming the Riemann Hypothesis,
it is still an approximate value to the real value count x for any
given prime (p).( pi(x)=p).

Granted, it does give the best approximation.

Indeed Riemann did give the best approximation, which is an
exact formula for pi(x) using all nontrivial zeroes of the Riemann
zeta function (assuming the Riemann hypothesis).

It is like trying to find a closed form for pi(x) which will never
happen.

Various exact "closed forms" are known, but pale in efficiency
with recursive techniques.

A good anology is the use of the gamma function for finding
the asymptotic value for (n!)

Gamma(n+1) = n! exactly.

It is sort of ironic that the primes and the factorials are related
because of Wilsons' theroem and both are not computable in
a closed form method but need a crutch such as the Riemann
Hypothesis for pi(x) and the gamma function for (n!) to give the
best approximation.

Dan

My point was that J K Haugland's answer (10) to your
question about the limit of the ratios of counts of primes
with successive lengths (in digits) is rigorously deducible
from the PNT.

regards, chip

Thanks chip, I can see that now.
Thanks to J K Haugland also.

This stuff is interesting where right now I am investigating
large composites that have two large prime factors of equal
digit length with a ratio ~ 2 between the two factors.
Very easy to factor using ECM. Digit length of composite is e+617.

Dan

.

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