The method of infinite ascent applied on $2^pA^6+B^3 = C^2$

Abstract

In this paper, we prove that for any positive integer $p$, when $p \equiv 1\pmod{6}$ or, $p \equiv 3 \pmod{6}$, the Diophantine equation: $2^p A^6 + B^3 = C^2$ has infinitely many co-prime integral solutions $A$, $B$, $C$. When $p=0$, this equation has only four integral solutions with $(A, B, C)= (\pm1,2,\pm3)$. For other integer values of $p$, the problem is open.