A question of Erd\"{o}s on $3$-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture

We describe how the Mordell-Weil group of rational points on a certain family of elliptic curves give rise to solutions to a conjecture of Erd\"{o}s on $3$-powerful numbers, and state a related conjecture which can be viewed as an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture.