A Note on the Diophantine Equation $lx^3 - kx^2 + kx - l = y^2$: The Cases $k=3l \pm 1$

Abstract

In this work, we investigate the Diophantine equation $lx^3-kx^2+kx-l = y^2$ where $k$ and $l$ are positive integers. The two results are Theorems 1.1 and 1.2. The first theorem states that if $k=3l-1$ and $l = \rho^2$, the above equation has a unique integer solution, namely $(x,y) = (1,0)$. The second theorem says that if $k=3l+1$ and $l\equiv 0,1,4,5,7 \pmod 8$ the above equation also has a unique solution, the pair $(x,y) = (1,0)$.