Bertrand's Postulate for Carmichael Numbers

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(log x/(log log x)2+delta) Carmichael numbers between x and x + x/(log x)(1/2+delta).