Bertrand's Postulate for Carmichael Numbers

TL;DR

This paper proves a Bertrand-like lower bound for the distribution of Carmichael numbers, showing they are sufficiently dense within certain intervals for large x, extending understanding of their distribution.

Contribution

It establishes a new lower bound on the number of Carmichael numbers within specific intervals, analogous to Bertrand's postulate for primes.

Findings

Carmichael numbers are densely distributed in large intervals.

A quantitative lower bound on Carmichael numbers in short intervals is proven.

The result extends the analogy between Carmichael numbers and prime distribution.

Abstract

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 $x+\frac{x}{(\log x)^{\frac{1}{2+\delta}}}$.