Complexity of Finding Values of the Generalized Taxicab Number

This short paper demonstrates a sketch of a generic algorithm that can generate integers written as the sum of N mth positive powers with any N≥2 and m≥1 in two different ways. As it is shown in the paper, such a generic algorithm is NP-hard. One can infer from this result that to generate exact values of the generalized "Taxicab" (m,N,j)– the smallest sum of N mth positive powers expressed in j different ways – is at least as hard as the hardest problems in NP. This implies that except for small N to find the generalized "Taxicab " (m,N,j) would be unfeasible on any computing device.