The Canadian Deuterium-Uranium (CANDU) pressurized heavy water reactor is fueled continuously at power, with alternate channels being fueled in opposite directions (continuous bidirectional fueling). The rate at which channels are refueled in various regions of the core determines the burnup distribution in the core. The burnup distribution in the core determines the power distribution. In present practice, the core is divided radially into two burnup regions having constant average discharge burnup. The limit on maximum neutron flux and the requirement for a critical system determine the size of the inner burnup region and the values of the burnups in the two regions. We can increase the core average exit burnup if we allow the burnup distribution to vary continuously rather than being regionwise constant. The purpose of this analysis is to derive an optimum burnup distribution that will maximize core average discharge burnup subject to a limit on maximum flux. This is equivalent to minimizing the total fuel feed rate. A set of equations describing the optimum distribution of burnup has been derived using calculus of variations techniques. These equations have been solved numerically in one-dimensional cylindrical geometry for homogeneous cores of approximately the size of current generation CANDU reactors.