On $A^ 4+B^ 4+C^ 4=D^ 4$

Abstract:

We use elliptic curves to find infinitely many solutions to ${A^4} + {B^4} + {C^4} = {D^4}$ in coprime natural numbers A, B, C, and D, starting with \[ {2682440^4} + {15365639^4} + {18796760^4} = {20615673^4}.\] We thus disprove the $n = 4$ case of Euler’s conjectured generalization of Fermat’s Last Theorem. We further show that the corresponding rational points $( \pm A/D, \pm B/D, \pm C/D)$ on the surface ${r^4} + {s^4} + {t^4} = 1$ are dense in the real locus. We also discuss the smallest solution, found subsequently by Roger Frye.