### Abstract

Let p _{1},⋯,p _{s} be distinct primes, and S the set of integers not divisible by primes different from p _{1},⋯, p _{s}. We give effectively computable upper bounds for n in the equations (1) f(x) = wy ^{n}, (5) F(x, z) = wy ^{n} and (9) ax ^{n} by ^{n} = c, where f ∈ Z[X] is a monic polynomial, F ∈ Z[X, Z] a monic binary form, the discriminants D(f), D(F) are contained in S, and x, y, z, w, a, b, c, n are unknown non-zero integers with z, w, a, b, c ∞ S, y ∉ S and n ≥ 3. It is a novelty in our paper that the upper bounds depend only on the product p _{1} ⋯ p _{s}, and, in case of (1) and (5), on deg f and deg F, respectively. The bounds are given explicitly in terms of p _{1} ⋯ p _{s}. Our results are established in more general forms, over an arbitrary algebraic number field. Equation (5) is reduced to an equation of type (9) over an appropriate extension of the ground field. The proofs involve among other things the best known estimates for linear forms in logarithms of algebraic numbers.

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

Pages (from-to) | 341-362 |

Number of pages | 22 |

Journal | Publicationes Mathematicae |

Volume | 65 |

Issue number | 3-4 |

Publication status | Published - 2004 |

### Fingerprint

### Keywords

- Binomial Thue-Mahler equations
- Discriminants of polynomials and binary forms
- S-unit equations
- Superelliptic equations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*65*(3-4), 341-362.

**Power values of polynomials and binomial Thue-Mahler equations.** / Györy, K.; Pink, István; Pintér, Ákos.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 65, no. 3-4, pp. 341-362.

}

TY - JOUR

T1 - Power values of polynomials and binomial Thue-Mahler equations

AU - Györy, K.

AU - Pink, István

AU - Pintér, Ákos

PY - 2004

Y1 - 2004

N2 - Let p 1,⋯,p s be distinct primes, and S the set of integers not divisible by primes different from p 1,⋯, p s. We give effectively computable upper bounds for n in the equations (1) f(x) = wy n, (5) F(x, z) = wy n and (9) ax n by n = c, where f ∈ Z[X] is a monic polynomial, F ∈ Z[X, Z] a monic binary form, the discriminants D(f), D(F) are contained in S, and x, y, z, w, a, b, c, n are unknown non-zero integers with z, w, a, b, c ∞ S, y ∉ S and n ≥ 3. It is a novelty in our paper that the upper bounds depend only on the product p 1 ⋯ p s, and, in case of (1) and (5), on deg f and deg F, respectively. The bounds are given explicitly in terms of p 1 ⋯ p s. Our results are established in more general forms, over an arbitrary algebraic number field. Equation (5) is reduced to an equation of type (9) over an appropriate extension of the ground field. The proofs involve among other things the best known estimates for linear forms in logarithms of algebraic numbers.

AB - Let p 1,⋯,p s be distinct primes, and S the set of integers not divisible by primes different from p 1,⋯, p s. We give effectively computable upper bounds for n in the equations (1) f(x) = wy n, (5) F(x, z) = wy n and (9) ax n by n = c, where f ∈ Z[X] is a monic polynomial, F ∈ Z[X, Z] a monic binary form, the discriminants D(f), D(F) are contained in S, and x, y, z, w, a, b, c, n are unknown non-zero integers with z, w, a, b, c ∞ S, y ∉ S and n ≥ 3. It is a novelty in our paper that the upper bounds depend only on the product p 1 ⋯ p s, and, in case of (1) and (5), on deg f and deg F, respectively. The bounds are given explicitly in terms of p 1 ⋯ p s. Our results are established in more general forms, over an arbitrary algebraic number field. Equation (5) is reduced to an equation of type (9) over an appropriate extension of the ground field. The proofs involve among other things the best known estimates for linear forms in logarithms of algebraic numbers.

KW - Binomial Thue-Mahler equations

KW - Discriminants of polynomials and binary forms

KW - S-unit equations

KW - Superelliptic equations

UR - http://www.scopus.com/inward/record.url?scp=11144322813&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=11144322813&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:11144322813

VL - 65

SP - 341

EP - 362

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 3-4

ER -