Seeds for every rational relation

Establish whether every rational relation k in e=a+k^3(b+c+d) carries a primitive seed satisfying a^4+b^4+c^4+d^4=e^4, equivalently whether an infinite subset of such relations carries elliptic curves arising from the construction.

Background

The paper defines a seed as a primitive solution of a4+b4+c4+d4=e4 satisfying the linear relation e=a+k3(b+c+d), with rational k. It exhibits seeds for eleven values of k and proves that each exhibited seed yields an infinite family through an associated elliptic curve.

The authors ask whether this phenomenon occurs for every relation k, and note that any seed would produce an infinite family by the same recipe. They also formulate the question in terms of proving that an infinite subset of relations carries elliptic curves.

References

Does every relation $k$ carry a seed? Any seed should give an infinite family by the same recipe (proof of Theorem~\ref{thm:main}). In other words, proving that an infinite subset of relations $k$ carries elliptic curves is an open problem.

— On $a^4+b^4+c^4+d^4=(a+27(b+c+d))^4$ and ten other new infinite families  (2610.12181 - Veselovac et al., 8 Oct 2026) in Section “Open questions,” first question

Does there exist a split into two quadric surfaces for some $k\ne1$?

— On $a^4+b^4+c^4+d^4=(a+27(b+c+d))^4$ and ten other new infinite families  (2610.12181 - Veselovac et al., 8 Oct 2026) in Section “Open questions,” second question