A Two-Tier Algebraic Schema to Map ${(A^4-C^4)/(D^4-B^4)}$ onto the Natural Numbers

Abstract

A brief history and two formulations of the Diophantine problem's requirements are presented. One tier consisting of three two-parameter solutions is studied for its ability to provide examples for the small natural numbers considered. Nested within it is a second tier consisting of five shifted-square solutions of the form $u^2+c$, where $u,c \in Q$. All told, they provide numerical examples for all but two $a \in N[1000]$, the set of natural numbers less than or equal to $1000$. A few open questions remain. Does this scheme of solutions cover every $a \in N[1000]$? If so, might they account for all $a \in N$? Are the three $tier_1$ solutions redundant with respect to the $a's$ they provide? Do other $tier_1$ and shifted-square $tier_2$ solutions exist?