A -multigrade equation is a Diophantine Equation of the form

for , ..., , where m and n are -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 where the minimum component of one of the vectors is 1.

Small-order examples are the (2, 3)-multigrade with and :

the (3, 4)-multigrade with and :

and the (4, 6)-multigrade with and :

A spectacular example with and is given by and (Guy 1994), which has sums

References

Chen, S. Equal Sums of Like Powers: On the Integer Solution of the Diophantine System.'' http://www.nease.net/~chin/eslp/

Gloden, A. Mehrgeradige Gleichungen. Groningen, Netherlands: Noordhoff, 1944.

Gloden, A. Sur la multigrade , , , , , , , , (, 3, 5, 7).'' Revista Euclides 8, 383-384, 1948.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 143, 1994.

Kraitchik, M. Multigrade.'' §3.10 in Mathematical Recreations. New York: W. W. Norton, p. 79, 1942.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 171-173, 1979.