Infinite families from nonlinear relations

Determine whether nonlinear relations among a,b,c,d,e support infinite families of primitive solutions to a^4+b^4+c^4+d^4=e^4 beyond the linear relations e=a+k^3(b+c+d) and the known relation a=b.

Background

The paper’s principal construction concerns linear relations of the form e=a+k3(b+c+d), for which known seeds lead to elliptic curves and infinite families. It also notes a separate family associated with the linear relation a=b.

The authors explicitly ask whether genuinely nonlinear relations among the five variables can likewise carry infinite families of solutions.

References

Do nonlinear relations in $a,b,c,d,e$ carry infinite families of eq:main? There is another set of infinite families on eq:main beside eq:slice, the relation $a=b$, due to a 2014 identity of Bremner, Choudhry and Ulas and a 2026 seed.

— 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,” third question