The Prime Wavelet Tree: Compressing the Carmichael Numbers

Abstract

We present two number-theoretic compression techniques. The first is \textit{local}, and compresses individual numbers. The second is \textit{global}, and compresses the entire set of numbers using a \textit{prime wavelet tree}, which is a data structure of independent interest. Our techniques apply to any set of numbers satisfying a Korselt-like criterion. We demonstrate our compression techniques using the recently completed tabulation of all $308,279,939$ Carmichael numbers less than $10^{24}$ that occupies $18.4$ gigabytes as text. When combined with standard compression techniques and a heuristic choice of divisors, the resulting file is $588.7$ megabytes ($31.2$ times smaller than the text file). The initial $80$-bit numbers now use about $15.3$ bits.

arXiv comment10 pages, 2 figures, 2 tables, prepared for DCC