NearSquarePrime 2009-06-12 21:23:03 2009-06-17 20:09:48 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 A {\em near-square prime} is a prime number $p$ of the form $n^2 + k$, with $n$ being any integer and $0 < |k| < |n|$ also an integer. Since for any nonzero real number $x$ it is always the case that $x^2 \geq 0$, it doesn't matter if $n$ is negative. \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|} 5 & & & & & & & & & & & & 149 \\ 4 & & & & & 29 & & 53 & & & & & \\ 3 & & & & & & & & 67 & & 103 & & \\ 2 & & & 11 & & & & & & 83 & & & \\ 1 & & 5 & & 17 & & 37 & & & & 101 & & \\ 0 & 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 121 & 144 \\ $-1$ & & 3 & & & & & & & & & & \\ $-2$ & & & 7 & & 23 & & 47 & & 79 & & & \\ $-3$ & & & & & & & & & & 97 & & \\ $-4$ & & & & & & & & & & & & \\ $-5$ & & & & & & 31 & & 59 & & & & 139 \\ \end{tabular} Fermat primes are near-square primes for $k = 1$ with the additional requirement that $n = 2^{2^m - 1}$, while Carol primes are near-square primes for $k = -2$ with the additional requirement that $n = 2^m - 1$. For $k = -1$, only $n = 2$ gives a prime, namely 3.