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 ,
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 50. Is anyone interested in pursuing a general proof?