- Mordell curve
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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
- ^ a b ""Mathworld Wolfram". October 17. http://mathworld.wolfram.com/MordellCurve.html.
Bibliography
- Louis Mordell, Diophantine Equations (1969)
Categories:- Algebraic curves
- Diophantine equations
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