### Abstract

We explicitly solve the equation Ax^{n}-By^{n} = ±1 and, along the way, we obtain new results for a collection of equations Ax^{n}-By^{n} = z^{m} with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≤ 3, AB = p^{α}q^{β} with nonnegative integers α and β and with primes 2 ≤ p <q <30. The proofs require a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.

Original language | English |
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Pages (from-to) | 61-77 |

Number of pages | 17 |

Journal | Fundamental and Applied Mathematics |

Volume | 16 |

Issue number | 5 |

Publication status | Published - 2010 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Applied Mathematics
- Geometry and Topology

### Cite this

*Fundamental and Applied Mathematics*,

*16*(5), 61-77.

**Binomial Thue equations, ternary equations, and power values of polynomials.** / Györy, K.; Pintér, A.

Research output: Contribution to journal › Article

*Fundamental and Applied Mathematics*, vol. 16, no. 5, pp. 61-77.

}

TY - JOUR

T1 - Binomial Thue equations, ternary equations, and power values of polynomials

AU - Györy, K.

AU - Pintér, A.

PY - 2010

Y1 - 2010

N2 - We explicitly solve the equation Axn-Byn = ±1 and, along the way, we obtain new results for a collection of equations Axn-Byn = zm with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≤ 3, AB = pαqβ with nonnegative integers α and β and with primes 2 ≤ p

AB - We explicitly solve the equation Axn-Byn = ±1 and, along the way, we obtain new results for a collection of equations Axn-Byn = zm with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≤ 3, AB = pαqβ with nonnegative integers α and β and with primes 2 ≤ p

UR - http://www.scopus.com/inward/record.url?scp=79959631454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959631454&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79959631454

VL - 16

SP - 61

EP - 77

JO - Fundamental and Applied Mathematics

JF - Fundamental and Applied Mathematics

SN - 1560-5159

IS - 5

ER -