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 infinity. 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).
If (x, y) is a rational point on a Mordell curve with y ≠ 0, then so is (x4 − 8nx/4y2, −x6 − 20nx3 + 8n2/8y3). Moreover, if xy≠ 0 and n is not 1 or −432, an infinite number of rational solutions can be generated this way. This formula is known as Bachet's duplication formula.[3]
^Gebel, J.; Pethö, A.; Zimmer, H. G. (1998). "On Mordell's equation". Compositio Mathematica. 110 (3): 335–367. doi:10.1023/A:1000281602647.
^Sequences OEIS: A081119 and OEIS: A081120.
^M. A. Bennett, A. Ghadermarzi (2015). "Mordell's equation : a classical approach" (PDF). LMS Journal of Computation and Mathematics. 18: 633–646. arXiv:1311.7077. doi:10.1112/S1461157015000182.
External links
J. Gebel, Data on Mordell's curves for –10000 ≤ n ≤ 10000
M. Bennett, Data on Mordell curves for –107 ≤ n ≤ 107