Quintic Reciprocity and Primality Test for Numbers of the Form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M~=~A5^{n} \pm ~\omega_{n}$\end{document}

The Quintic Reciprocity Law is used to produce an algorithm, that runs in polynomial time, and that determines the primality of numbers M such that M4 − 1 is divisible by a power of 5 which is larger that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt{M}$\end{document}, provided that a small prime p, p ≡ 1 (mod 5) is given, such that M is not a fifth power modulo p. The same test equations are used for all such M. If M is a fifth power modulo p, a sufficient condition that determines the primality of M is given.