Regeneration of the magnetic field by convection in the core places demands on heat flow into the base of the mantle. If the heat flow is too low, thermal convection is shut off and the rate of generation of compositional buoyancy by the solidification of the core becomes too low to sustain the geodynamo. Conversely, a large heat flow causes rapid growth of the inner core, so that convection prior to the appearance of the inner core must be sustained by thermal buoyancy alone. The attendant requirements on primordial heat become more severe as the age of the inner core decreases. We show that estimates of the present-day heat flow satisfy the power requirements for the geodynamo. However, the resulting thermal history is incompatible with estimates of mantle temperatures prior to 3 Ga. This discrepancy can be resolved by accumulating radioactive isotopes in D″ or adding heat sources to the core.