Alford et al. [1] proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand’s postulate could be proven for Carmichael numbers. In this paper, we answer this question, proving the stronger statement that for all $\delta>0$ and $x$ sufficiently large in terms of $\delta $, there exist at least $e^{\frac {\log x}{(\log \log x)^{2+\delta }}}$ Carmichael numbers between $x$ and $x+\frac {x}{(\log x)^{\frac {1}{2+\delta }}}$.

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卡迈克尔数的伯特兰公设

奥尔福德等人。[1] 证明有无限多个卡迈克尔数。在同一篇论文中，他们询问是否可以为卡迈克尔数证明类似于伯特兰假设的陈述。在本文中，我们回答了这个问题，证明了对于所有的 $\delta>0$ 和 $x$ 就 $\delta $ 而言足够大，至少存在 $e^{\frac {\log x }{(\log \log x)^{2+\delta }}}$ 在$x$ 和$x+之间的卡迈克尔数\frac {x}{(\log x)^{\frac {1}{2+\增量}}}$。