### Abstract

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals ℚ. Here we illustrate our method by applying it to Mordell's Equation y^{2} = x^{3} + k for 0 ≠ k ∈ ℤ and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in ℤ for all integers k within the range 0 <|k| ≤ 10 000 and partially extend the computations to 0 <|k| ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C = 5. Some other interesting observations are made concerning large integer points, large generators of the Mordell-Weil group and large Tate-Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

Original language | English |
---|---|

Pages (from-to) | 335-367 |

Number of pages | 33 |

Journal | Compositio Mathematica |

Volume | 110 |

Issue number | 3 |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

- Birch & Swinnerton-Dyer conjecture
- Elliptic curve
- Elliptic logarithm
- Height
- Integral points
- L-series
- Mordell-Weil group
- Rank
- Regulator
- Torsion

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

*110*(3), 335-367.

**On Mordell's equation.** / Gebel, J.; Pethő, A.; Zimmer, H. G.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 110, no. 3, pp. 335-367.

}

TY - JOUR

T1 - On Mordell's equation

AU - Gebel, J.

AU - Pethő, A.

AU - Zimmer, H. G.

PY - 1998

Y1 - 1998

N2 - In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals ℚ. Here we illustrate our method by applying it to Mordell's Equation y2 = x3 + k for 0 ≠ k ∈ ℤ and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in ℤ for all integers k within the range 0 <|k| ≤ 10 000 and partially extend the computations to 0 <|k| ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C = 5. Some other interesting observations are made concerning large integer points, large generators of the Mordell-Weil group and large Tate-Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

AB - In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals ℚ. Here we illustrate our method by applying it to Mordell's Equation y2 = x3 + k for 0 ≠ k ∈ ℤ and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in ℤ for all integers k within the range 0 <|k| ≤ 10 000 and partially extend the computations to 0 <|k| ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C = 5. Some other interesting observations are made concerning large integer points, large generators of the Mordell-Weil group and large Tate-Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

KW - Birch & Swinnerton-Dyer conjecture

KW - Elliptic curve

KW - Elliptic logarithm

KW - Height

KW - Integral points

KW - L-series

KW - Mordell-Weil group

KW - Rank

KW - Regulator

KW - Torsion

UR - http://www.scopus.com/inward/record.url?scp=0000253128&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000253128&partnerID=8YFLogxK

M3 - Article

VL - 110

SP - 335

EP - 367

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 3

ER -