This is more of a curiosity than a research question, but I could not find it answered anywhere. What is the largest $N$ for which the statement in the title is true? I have recently read that the largest known prime is $2^{82589933} − 1$, but I imagine it is not known if, for example, $2^{82589933} − 3$ is prime. A closely related question: if $\pi(x)$ is the prime counting function, that is, $\pi(x)$ is the number of primes not exceeding $x$, what is the largest value of $n$ for which $\pi(n)$ is known exactly?

## 1 Answer

As explained in the comments, there is no well-defined answer to this question, because there is no organized effort to test primality of all numbers up to a certain bound.

However, there is a (more or less) well-defined answer (at any given point in time) to a related question: What is the largest $N$ for which the primality status of $2^n - 1$ is known for all $n\le N$? The answer, as of this writing, is 57885161 (though some "insiders" in the GIMPS project may be able to report a slightly higher number than that—but not higher than 74207281).

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