  In this article, we make use of some known method to investigate some
properties of the numbers represented as sums of two equal odd powers, i.e.,
the equation $x^n+y^n=N$ for $n\ge3$. It was originated in developing
algorithms to search new taxicab numbers (i.e., naturals that can be
represented as a sum of positive cubes in many different ways) and to verify
their minimality. We discuss properties of diophantine equations that can be
used for our investigations. This techniques is applied to develop an algorithm
allowing us to compute new taxicab numbers (i.e., numbers represented as sums
of two positive cubes in $k$ different ways), for $k=7...14$.
