The Mathematics of the Mordell Equation

The Mordell equation is y² = x³ + k. For k = 17? Let’s try k = 17: find integer solutions. The equation is a classic example of an elliptic curve. For many k, there are finitely many integer solutions. For k=17, one solution is (x,y) = (−2, ±3)? (−2)³+17 = −8+17=9, y=±3. Also (−1, ±4)? −1+17=16, y=±4. Also (2, ±5)? 8+17=25, y=±5. Actually (2,5) works, (4, ±9)? 64+17=81, y=±9. Also (8, ±23)? 512+17=529=23². The Mordell equation has infinitely many integer solutions for some k (e.g., k=0: y²=x³ has trivial families). It is a gateway to the study of rational points on elliptic curves.