A (k,d)-multigrade is a diophantine equation of the form

, that is valid for all powers

. We can call k, the highest valid exponent, the "level" and d, the dimension of each vector

the "order." Consider altering the multigrade by another vector

as such:

, provided

for all i. In another thread, Wonderboy1953 conjectured that for the vector

, no multigrade exists such that its altered version holds on any level greater than 2. That is, given a (k,d)-multigrade

,

. This is equivalent to the following formulation using matrices...

Find two vectors

,

Define

Define

and

Such that

. In general, A and B are independent of each other, so the nullspace of matrix M has at least dimension 2, requiring

.

In search of a counterexample, I have verified that when

, no d=5 solutions exist for vectors

with components less than 40. Is anyone interested in pursuing a general proof?