We now have sufficient machinery at our disposal to tackle Fermat’s Last Theorem in a special case, namely when the exponent n in the equation xn + yn = zn is a so-called ‘regular prime’, and when n does not divide any of x, y, or z. We begin with a short historical survey to set this version of the problem in perspective. Following this we show how elementary methods dispose of the case n = 4 and reduce the problem to odd prime values of n. In this chapter we do not deal with the case where one of x, y, or z is divisible by n, neither do we deal with irregular prime n. These cases are described in Chapter 14. In a final discursive section we discuss the regularity property and some related matters.