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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?

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    $\begingroup$ 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. $\endgroup$
    – Community Bot
    Jan 23 at 3:47
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    $\begingroup$ Is $e$ meant to be the base of natural logarithms? $\endgroup$ Jan 23 at 3:48
  • $\begingroup$ 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. $\endgroup$
    – LSpice
    Jan 23 at 4:20
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    $\begingroup$ @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. $\endgroup$
    – Will Sawin
    Jan 23 at 4:33
  • $\begingroup$ 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. $\endgroup$
    – Yinpo
    Jan 23 at 15:57

1 Answer 1

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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.

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  • $\begingroup$ "for all irrational algebraic integers" I'm assuming you meant "real numbers" rather than "integers" here? $\endgroup$
    – Stef
    Jan 23 at 13:57
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    $\begingroup$ @Stef No, I mean an algebraic integer (a root of a monic polynomial with integer coefficients) that is irrational. $\endgroup$
    – 2734364041
    Jan 23 at 14:01

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