A (k,l)-multigrade equation is a Diophantine equation of the form sum_(i=1)^ln_i^j=sum_(i=1)^lm_i^j (1) 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 (Madachy 1979), so multigrades can always be put in a form w…