On 𝐴⁴+𝐵⁴+𝐶⁴=𝐷⁴

Source abstract

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