Complete solutions to a family of cubic Diophantine equations

Abstract The following theorem is proved: If n ≥ 1.365 × 107, then the Diophantine equation x 3 −(n−1)x 2 y−(n+2)xy 2 −y 3 =±1 has only the “trivial” solutions (±1,0), (0,±1), (±1,∓1) . Moreover, we show that for 0 ≤ n ≤ 1000, (∗) has non-trivial solutions if, and only if, n = 0, 1, 3. Finally, we conjecture that if n > 3, then (∗) has only trivial solutions.