Number primes digits

The prime number theorem will give an approximate answer. You want the number of primes between $10^{99}$ and $10^{100}$ For the simplest approximation, this is just $$\frac{10^{100}}{\log 10^{100}}-\frac{10^{99}}{\log 10^{99}}=\frac{10^{100}}{100\log 10}-\frac{10^{99}}{99\log 10}=10^{99}\left(\frac {10}{\log 100}-\frac 1{\log 99}\right)$$

If this is too big for your calculator, it is. Wolfram alpha will give the answer as approximately $3.9\cdot 10^{97}$. You can use the more accurate formula for the prime number in terms of the logarithmic integral if you want, but the issues are the same.


To determine the number of primes with $100$ digits exactly, you would need to calculate, at least effectively, the value of $\pi\left(10^{100}\right) - \pi\left(10^{99}\right)$, where $\pi(x)$ is the count of # of primes up to $x$. However, regarding determining $\pi\left(10^{n}\right)$ for larger values of $n$, according to Prime-counting function,

The value for $10^{27}$ was published in $2015$ by David Baugh and Kim Walisch.

Since the number of primes up to $10^{27}$ was only calculated about $4$ years ago, I highly doubt there would be anything for either $10^{99}$ or $10^{100}$ yet.

Nonetheless, there are various estimates for the # of primes in a larger interval which give considerably more accurate values than you would get by just using the Prime Number Theorem. For example, the Inequalities section of Wikipedia's "Prime-counting function" article gives several good ones you can use instead. Also, the upper & lower bounds used in the inequalities will allow you to determine a maximum error of the estimate you get when you use one of them.

Update: This answer gives an approximation for the # of primes using the Prime Number Theorem formula. Although Wolfram Alpha can calculate it directly, you can also actually fairly easily factor out $10^{99}$, with this allowing any calculator or program which supports natural logarithms to do the calculations of the remaining parts. In particular, you get

$$\begin{equation}\begin{aligned} \pi\left(10^{100}\right) - \pi\left(10^{99}\right) & \approx \frac{10^{100}}{100\ln 10} - \frac{10^{99}}{99\ln 10} \\ & = 10^{99}\left(\frac{10}{100\ln 10} - \frac{1}{99\ln 10}\right) \\ & = 10^{99}\left(\frac{1}{\ln 10}\right)\left(\frac{1}{10} - \frac{1}{99}\right) \\ & \approx 10^{99} \times 0.0390426 \\ & = 3.90426 \times 10^{97} \end{aligned}\end{equation}\tag{1}\label{eq1A}$$

Note I used the Windows $7$ calculator program to do the calculations to get the final answer shown above.

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Number Theory

Prime Numbers

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