A (k, l)-multigrade equation is a Diophantine equation of the form sum_(i = 1)^l n_i^j = sum_(i = 1)^l m_i^j for j = 1, ..., k, where m and n are l-vectors. Multigrade identities remain valid if a constant is added to each element of m and n, so multigrades can always be put in a form where the minimum component of one of the vectors is 1. Moessner and Gloden give a bevy of multigrade equations.

Diophantine equation | Prouhet-Tarry-Escott problem