Integers that are sums of two rational sixth powers. Issue 1 (7th March 2023)

• Record Type: Journal Article
• Authors: Newton, Alexis; Rouse, Jeremy
• Is Part Of: Canadian mathematical bulletin =. Volume 66:Issue 1(2023)
• Page Start: 166
• Page End: 177
• Publication Date: 2023-03-07
• Subjects: 11G05 -- 14H45 -- 11Y50; Elliptic curve -- Mordell–Weil sieve -- Fermat curve -- sixth power; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439522000157
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: We prove that $164\, 634\, 913$ is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell–Weil sieve, to rule out the existence of rational points on $C_{k}$ for various k .