RedmondSunConjecture 2007-08-02 18:57:52 2007-08-19 16:13:20 Conjecture % 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 Conjecture. (Stephen Redmond \& Zhi-Wei Sun) Given positive integers $x$ and $y$, and exponents $a$ and $b$ (with all these numbers being greater than 1), if $x^a \neq y^b$, then between $x^a$ and $y^b$ there are always primes, with only the following ten exceptions: \begin{enumerate} \item There are no primes between $2^3$ and $3^2$. \item There are no primes between $5^2$ and $3^3$. \item There are no primes between $2^5$ and $6^2$. \item There are no primes between $11^2$ and $5^3$. \item There are no primes between $3^7$ and $13^3$. \item There are no primes between $5^5$ and $56^2$. \item There are no primes between $181^2$ and $2^{15}$. \item There are no primes between $43^3$ and $282^2$. \item There are no primes between $46^3$ and $312^2$. \item There are no primes between $22434^2$ and $55^5$. \end{enumerate} See A116086 in Sloane's OEIS for a listing of the perfect powers beginning primeless ranges before the next perfect power. As of 2007, no further counterexamples have been found past $55^5$.