Mordell curve

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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).

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

In 1998, J. Gebel, A. Pethö, H. G. Zimmer found all integers points for 0 < |n| ≤ 104.[3][4]

In 2015, M. A. Bennett and A. Ghadermarzi computed integer points for 0 < |n| ≤ 107.[5]

List of solutions[]

The following is a list of solutions to the Mordell curve for |n| ≤ 25. Only solutions with y ≥ 0 are shown.

n (x,y)
1 (−1, 0), (0, 1), (2, 3)
2 (−1, 1)
3 (1, 2)
4 (0, 2)
5 (−1, 2)
6 –
7 –
8 (−2, 0), (1, 3), (2, 4), (46, 312)
9 (−2, 1), (0, 3), (3, 6), (6, 15), (40, 253)
10 (−1, 3)
11 –
12 (−2, 2), (13, 47)
13 –
14 –
15 (1, 4), (109, 1138)
16 (0, 4)
17 (−1, 4), (−2, 3), (2, 5), (4, 9), (8, 23), (43, 282), (52, 375), (5234, 378661)
18 (7, 19)
19 (5, 12)
20 –
21 –
22 (3, 7)
23 –
24 (−2, 4), (1, 5), (10, 32), (8158, 736844)
25 (0, 5)
n (x,y)
−1 (1, 0)
−2 (3, 5)
−3 –
−4 (5, 11), (2, 2)
−5 –
−6 –
−7 (2, 1), (32, 181)
−8 (2, 0)
−9 –
−10 –
−11 (3, 4), (15, 58)
−12 –
−13 (17, 70)
−14 –
−15 (4, 7)
−16 –
−17 –
−18 (3, 3)
−19 (7, 18)
−20 (6, 14)
−21 –
−22 –
−23 (3, 2)
−24 –
−25 (5, 10)

References[]

  1. ^ a b Weisstein, Eric W. "Mordell Curve". MathWorld.
  2. ^ Louis Mordell (1969). Diophantine Equations.
  3. ^ Gebel, J.; Pethö, A.; Zimmer, H. G. (1998). "On Mordell's equation". Compositio Mathematica. 110 (3): 335–367. doi:10.1023/A:1000281602647.
  4. ^ Sequences OEIS: A081119 and OEIS: A081120.
  5. ^ 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[]

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