Mordell curve

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 an integer and where n ≠ 0[1]. In Mordell curves, if (x, y) is a solution, it therefore follows that (x, -y) is as well.

These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He showed that for n fixed there are only finitely many solutions (x, y) in integers.

There are certain values for n in a Mordell curve in which the equation gives no integer solutions, these values are 6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42...[1]

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.

References

  1. ^ a b ""Mathworld Wolfram". October 17. http://mathworld.wolfram.com/MordellCurve.html. 

Bibliography


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