Tweeting Primes

I recently discovered the Twitter account @primes. Every hour, they tweet the subsequent prime number. This made me wonder two things. First, what is the largest prime that you can tweet (in base-10 encoding in 140 characters).1 Second, how long until they get there.

Doing some quick calculations in Mathematica, I believe the largest 140 digit prime is the following:


Wolfram Alpha confirms that this is prime and that the next prime is 141 characters.

As for how long it would take, recall that the number of primes less than $n$ is approximately $\frac{n}{\ln n}$. The number of primes less than $10^{141}$ is approximately

$$\pi(10^{140}) = \frac{10^{140}}{140\cdot \ln 10} = 3.1\cdot 10^{137}.$$

That’s $3\cdot 10^{57}$ times the estimated number of atoms in the universe. Looks like @primes should be able to tweet for a while.

  1. The largest known prime is $2^{57,885,161} − 1$ and has 17,425,170 digits. ↩︎

Last updated on Feb 05, 2024 16:42 -0500