Coverage of a ≤ 1000 by the two-tier schema

Determine whether the two-tier algebraic schema—comprising the first-tier two-parameter nests A, B1, and B2 (Table 1) together with the second-tier shifted-square nests (Table 2)—produces at least one solution to the quartic Diophantine equation A^4 + a B^4 = C^4 + a D^4 for every natural number a ≤ 1000.

Background

The paper proposes a two-tier algebraic schema of parametric nests that generate solutions to A^4 + a B^4 = C^4 + a D^4. Extensive computations covered nearly all a in N[1000], leaving a small fallout subset.

The authors explicitly raise whether their scheme achieves complete coverage within this bounded range, framing it as an open question tied to their computational findings.