Need a check on calculating prime distributions for large values

[mesa]

My calculator isn't at all happy running the likely hood of finding a prime at 10,000 digits. Since there is a correlation very close to 1/2 the number of primes for each increase of 1000 digits after 1000 digits I was thinking I could just use,

1/2^(n/1000)×1151.3 = probability of finding a prime for n # of digits

This doesn't work at all for a small number of digits but I am only concerned about gigantic primes and above. Anyone have an idea to about how many digits this function will be accurate?

 

[mfb]

Prime number theorem
For numbers around a large number N, approximately 1/ln(N) of those numbers are prime.

For numbers with 10000 digits, the fraction of primes is $\frac{1}{\ln(10^{10000}) } \approx \frac{1}{23000}$

 

[mesa]

Prime number theorem
For numbers around a large number N, approximately 1/ln(N) of those numbers are prime.

For numbers with 10000 digits, the fraction of primes is $\frac{1}{\ln(10^{10000}) } \approx \frac{1}{23000}$

Okay so the function I wrote is bunk :)

Where did I get a calculator that can handle the ln10^10000?

 

[mfb]

WolframAlpha can do it

Alternatively, use log rules: ln(10¹⁰⁰⁰⁰)=10000*ln(10).

 

[mesa]

WolframAlpha can do it

It sure can, thanks for the link!

Alternatively, use log rules: ln(10¹⁰⁰⁰⁰)=10000*ln(10).

Realized that right after I left to get my kids. Brain back-logged :)