# Are there infinitely many primes of the form $\lfloor e x\rfloor$ for $x\in\mathbb{Z}^+$?

Construct a function $$f(x)=\lfloor e x\rfloor$$. For each positive integer $$x$$, $$f(x)$$ will be a positive integer. Among these integers $$f(x)$$, are there an infinite number of primes?

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Jan 23 at 3:47
• Is $e$ meant to be the base of natural logarithms? Jan 23 at 3:48
• I made an attempt to clarify the wording, but did not explicitly venture an assumption whether, as in @MichaelEngelhardt's comment, $\ln(e) = 1$. Hopefully @‍Yinpo will clarify, although @‍2734364041's answer suggests it doesn't matter. Jan 23 at 4:20
• @LSpice It does matter in that one wants $e$ to be an irrational number for the result to be applied - e.g. if $e=2$ there is a problem. Jan 23 at 4:33
• I mean 𝑒 is an example of an irritational number, the base of natural logarithms. Or generalized speaking, this could be random irrational number. Sorry for the ambiguity of my explanation. Jan 23 at 15:57

Something is known in considerably greater generality. Let $$\lVert x\rVert=\min_{n\in\mathbb{Z}}|x-n|$$. For $$\gamma\in\mathbb{R}-\mathbb{Q}$$, define
$$\tau(\gamma) = \sup\Big\{\rho\in\mathbb{R}\colon \liminf_{n\to\infty} n^{\rho}\lVert\gamma n\rVert=0\Big\},$$
where the limit inferior is as $$n\to\infty$$ along the integers. Dirichlet's approximation theorem implies that for all irrational $$\gamma$$, we have $$\tau(\gamma)\geq 1$$. Khinchin and Roth proved that $$\tau(\gamma)=1$$ for almost all real numbers $$\gamma$$ (in the sense of Lebesgue measure), and $$\tau(\gamma)=1$$ for all (real) irrational algebraic integers $$\gamma$$.
If $$\gamma>1$$ and $$\tau(\gamma)<\infty$$, then there is an asymptotic prime number theorem that counts the primes of the form $$p=\lfloor \gamma n+\beta\rfloor$$, where $$n\geq 1$$ is an integer and $$\beta\in\mathbb{R}$$ is fixed. A standard source for this is the paper Prime numbers with Beatty sequences by Banks and Shparlinski.