On the equation x 4 - y 4 = D

Abstract Let k be a perfect field. Assuming that the Mordell–Weil groups of certain elliptic curves are Z / 2 Z $\mathbb{Z}/2\mathbb{Z}$ , we find solutions of the equation x 4 − y 4 = D in all cubic extensions of k . This significantly extends the result of Tho in [“On the difference of two fourth powers,” Proc. Edinb. Math. Soc., II. Ser. , vol. 67, no. 1, pp. 142–150, 2024]. As an application, for prime p ≡ 3 (mod 8), we completely solve the equation x 4 − y 4 = p n in all cyclic cubic number fields.