Thabit number ThabitNumber 2006-04-29 14:53:13 2007-07-11 15:55:42 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here An integer of the form $3 \cdot 2^n - 1$, or $2^{n + 1} + 2^n - 1$. They are listed in A055010 of Sloane's OEIS. The Thabit numbers are a subset of the Proth numbers. The mathematician and astronomer Thabit ibn Qurra studied these numbers in search of a formula for amicable pairs. He found that when two consecutive Thabit numbers are also prime numbers (corresponding to indices $n$ and $n - 1$) and $9 \cdot 2^{2n - 1} - 1$ is a prime number, too, then these numbers multiplied by $2^n$ will reveal an amicable pair. The only $n$ known to fit these criteria are 2, 4 and 7. The largest Thabit number known to be prime corresponds to index 2312734, its immediate lower neighbor is composite. It is conjectured that the nimfactorial of a Thabit number is always 2.