Mordell curve

y² = x³ + 1, with solutions at (-1, 0), (0, 1) and (0, -1)

In algebra, a Mordell curve is an elliptic curve of the form y² = x³ + n, where n is a fixed non-zero integer.^[1]

These curves were closely studied by Louis Mordell,^[2] from the point of view of determining their integer points. He showed that every Mordell curve contains only finitely many integer points (x, y). In other words, the differences of perfect squares and perfect cubes tend to ∞. The question of how fast was dealt with in principle by Baker's method. Hypothetically this issue is dealt with by Marshall Hall's conjecture.

Properties

If (x, y) is an integer point on a Mordell curve, then so is (x, -y).

There are certain values of n for which the corresponding Mordell curve has no integer solutions;^[1] these values are:

6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... (sequence A054504 in OEIS).

-3, -5, -6, -9, -10, -12, -14, -16, -17, -21, -22, ... (sequence A081121 in OEIS).

In 1998, J. Gebel, A. Pethö, H. G. Zimmer found all integers solutions for 0 < |n| ≤ 10⁴.^[3] (Data on Mordell's curves for –10000 ≤ n ≤ 10000, A081119, A081120).

Example

Fermat proved that the only solutions of $m^2+2=n^3$ are $n=3,m=\pm 5$.

References

1. Weisstein, Eric W., "Mordell Curve", MathWorld.
2. Louis Mordell (1969). Diophantine Equations.
3. http://link.springer.com/article/10.1023%2FA%3A1000281602647

External links

• Data on Mordell's curves for –10000 ≤ n ≤ 10000