### Abstract

Using a method due to E. Thomas, we prove that if a> 9.9 10^{27}then the Diophantine equations. x^{4}-x^{3}y-x^{2}y^{2}+ axy^{3}+y^{4}=1 and x^{4}-ax^{3}y-3x^{2}y^{2}+ axy^{3}+y^{4}=±1 have exactly twelve solutions, namely (x, y) = (0, ±1), (±1, 0), (±1, ±1), (+-1, ±1), (±a, ±l), (±\, Ta) and eight solutions, (x, y) = (0, ±1), (±1, 0), (±1, ±1), (±1, Tl), respectively.

Original language | English |
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Pages (from-to) | 777-798 |

Number of pages | 22 |

Journal | Mathematics of Computation |

Volume | 57 |

Issue number | 196 |

DOIs | |

Publication status | Published - 1991 |

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### Keywords

- Linear forms in the logarithms of algebraic numbers
- Thue equation

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

**Complete solutions to families of quartic thue equations.** / Pethő, A.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 57, no. 196, pp. 777-798. https://doi.org/10.1090/S0025-5718-1991-1094956-7

}

TY - JOUR

T1 - Complete solutions to families of quartic thue equations

AU - Pethő, A.

PY - 1991

Y1 - 1991

N2 - Using a method due to E. Thomas, we prove that if a> 9.9 1027then the Diophantine equations. x4-x3y-x2y2+ axy3+y4=1 and x4-ax3y-3x2y2+ axy3+y4=±1 have exactly twelve solutions, namely (x, y) = (0, ±1), (±1, 0), (±1, ±1), (+-1, ±1), (±a, ±l), (±\, Ta) and eight solutions, (x, y) = (0, ±1), (±1, 0), (±1, ±1), (±1, Tl), respectively.

AB - Using a method due to E. Thomas, we prove that if a> 9.9 1027then the Diophantine equations. x4-x3y-x2y2+ axy3+y4=1 and x4-ax3y-3x2y2+ axy3+y4=±1 have exactly twelve solutions, namely (x, y) = (0, ±1), (±1, 0), (±1, ±1), (+-1, ±1), (±a, ±l), (±\, Ta) and eight solutions, (x, y) = (0, ±1), (±1, 0), (±1, ±1), (±1, Tl), respectively.

KW - Linear forms in the logarithms of algebraic numbers

KW - Thue equation

UR - http://www.scopus.com/inward/record.url?scp=84966240105&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966240105&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1991-1094956-7

DO - 10.1090/S0025-5718-1991-1094956-7

M3 - Article

AN - SCOPUS:84966240105

VL - 57

SP - 777

EP - 798

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 196

ER -