How many primes are there in a certain range of numbers?

[newcool]

Hi, what would be the best estimate in the # of primes between $$10^{100}$$ and $$10^{101}$$

thanks

 

[kreil]

I'm not sure if there is a newer equation, there probably is one from riemman, but Gauss had a formula for approximating the number of primes up to any number x:

$$Li(x)=\int_0^x\frac{dt}{log(t)}$$

You could compute this for 10¹⁰⁰ and then for 10¹⁰¹ and subtract the first result from the second and it will give a good estimate.

 

[tacman]

The answer to this question depends on the specific range of numbers and the method used to count primes. However, based on the Prime Number Theorem, we can estimate the number of primes in a given range by using the following formula:

N/log(N) where N is the largest number in the range.

In this case, N = 10^101, so the estimated number of primes in the range of 10^100 to 10^101 would be approximately 10^101/log(10^101) = 4.34 x 10^98.

This is just an estimate and the actual number of primes could be slightly different. Additionally, if you are looking for a more precise calculation, you can use a prime counting function or a computer algorithm to count the primes in the given range.