Existence of smaller rational points on the quartic curves

Determine whether the quartic curves constructed from the listed seeds possess rational points with smaller height than the corresponding seed points, thereby yielding smaller primitive solutions of the equation a^4+b^4+c^4+d^4=e^4 subject to the relations e=a+k^3(b+c+d).

Background

For each of the eleven displayed values of k, the paper constructs an elliptic quartic w2=Q(r) from a known seed and proves that the seed determines a rational point of infinite order. Multiples of that point generate infinitely many rational points and, via the paper’s substitutions, infinitely many primitive solutions of the four-fourth-powers equation.

The unresolved issue is whether any rational point on one of these quartics occurs before the point corresponding to the known seed in the relevant size ordering. Such a point would produce a smaller primitive solution than the seed used to construct the infinite family.

References

Whether the quartics have rational points smaller than their seeds, and hence smaller primitive solutions, is openquestion~VI.

— 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 “Infinitely many primitive solutions,” final paragraph before Section “Seed discovery”