Alford, Granville, and Pomerance 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 $x+\frac{ x}{ (\log x)^{ \frac{ 1}{ 2+\delta}}}$.
Categories: math.NT