  It is a classical problem in algebraic number theory to decide if a number
field admits power integral bases and further to calculate all generators of
power integral bases. This problem is especially delicate to consider in an
infinite parametric family of number fields. In the present paper we
investigate power integral bases in the infinite parametric family of pure
quartic fields $K=Q(\sqrt[4]{m})$.
  We often pointed out close connection of various types of Thue equations with
calculating power integral bases. In this paper we determine power integral
bases in pure quartic fields by reducing the corresponding index form equation
to quartic binomial Thue equations. We apply recent results on the solutions of
binomial Thue equations of type $a^4-mb^4=\pm 1\; ({\rm in}\; a,b\in Z)$ as
well as we perform an extensive calculation by a high performance computer to
determine "small" solutions (with $\max(|a|,|b|)<10^{500}$) of binomial Thue
equations for $m<10^7$.
  Using these results on binomial Thue equations we characterize power integral
bases in infinite subfamilies of pure quartic fields. For $1< m< 10^7$ we give
all generators of power integral bases with coefficients $<10^{1000}$ in the
integral basis.
