### Abstract

Abstract We prove that for any positive integers x,d and k with gcd (x,d)=1 and 311, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

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

Pages (from-to) | 845-864 |

Number of pages | 20 |

Journal | Compositio Mathematica |

Volume | 145 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2009 |

### Fingerprint

### Keywords

- arithmetic progression
- modular forms
- perfect powers
- ternary diophantine equations

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

*145*(4), 845-864. https://doi.org/10.1112/S0010437X09004114

**Perfect powers from products of consecutive terms in arithmetic progression.** / Györy, K.; Hajdu, L.; Pintér, Á.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 145, no. 4, pp. 845-864. https://doi.org/10.1112/S0010437X09004114

}

TY - JOUR

T1 - Perfect powers from products of consecutive terms in arithmetic progression

AU - Györy, K.

AU - Hajdu, L.

AU - Pintér, Á

PY - 2009/7

Y1 - 2009/7

N2 - Abstract We prove that for any positive integers x,d and k with gcd (x,d)=1 and 311, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

AB - Abstract We prove that for any positive integers x,d and k with gcd (x,d)=1 and 311, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

KW - arithmetic progression

KW - modular forms

KW - perfect powers

KW - ternary diophantine equations

UR - http://www.scopus.com/inward/record.url?scp=77957230672&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957230672&partnerID=8YFLogxK

U2 - 10.1112/S0010437X09004114

DO - 10.1112/S0010437X09004114

M3 - Article

AN - SCOPUS:77957230672

VL - 145

SP - 845

EP - 864

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 4

ER -