Mordell curve
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).
The specific case where n = −2 is also known as Fermat's Sandwich Theorem.[3]
List of solutions
The following is a list of solutions to the Mordell curve y2 = x3 + n 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) |
In 1998, J. Gebel, A. Pethö, H. G. Zimmer found all integers points for 0 < |n| ≤ 104.[4][5]
In 2015, M. A. Bennett and A. Ghadermarzi computed integer points for 0 < |n| ≤ 107.[6]
References
- ^ a b Weisstein, Eric W. "Mordell Curve". MathWorld.
- ^ Louis Mordell (1969). Diophantine Equations.
- ^ Weisstein, Eric W. "Fermat's Sandwich Theorem". MathWorld. Retrieved 24 March 2022.
- ^ 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