Elkies, Noam2009-05-121999Elkies, Noam D. The Klein quartic in number theory. In The Eightfold Way: The Beauty of Klein's Quartic Curve, ed. Sylvio Levi, 51-102. Mathematical Sciences Research Institute publications, 35. Cambridge: Cambridge University Press.http://nrs.harvard.edu/urn-3:HUL.InstRepos:2920120We describe the Klein quartic <i>X</i> and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of <i>X</i>; an explicit identification of with the modular curve X(7); and applications to the class number 1 problem and the case <i>n</i>= 7 of Fermat.en-US