### Abstract

The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k ≤ 5, under the assumption that P(b) ≤ p _{k} , the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.

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

Pages (from-to) | 19-33 |

Number of pages | 15 |

Journal | Monatshefte fur Mathematik |

Volume | 145 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 2005 |

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### Keywords

- Binomial Thue equations
- Exponential diophantine equations
- Product of consecutive integers

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Monatshefte fur Mathematik*,

*145*(1), 19-33. https://doi.org/10.1007/s00605-004-0278-8

**Almost perfect powers in products of consecutive integers.** / Györy, K.; Pintér, Á.

Research output: Contribution to journal › Article

*Monatshefte fur Mathematik*, vol. 145, no. 1, pp. 19-33. https://doi.org/10.1007/s00605-004-0278-8

}

TY - JOUR

T1 - Almost perfect powers in products of consecutive integers

AU - Györy, K.

AU - Pintér, Á

PY - 2005/5

Y1 - 2005/5

N2 - The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k ≤ 5, under the assumption that P(b) ≤ p k , the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.

AB - The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k ≤ 5, under the assumption that P(b) ≤ p k , the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.

KW - Binomial Thue equations

KW - Exponential diophantine equations

KW - Product of consecutive integers

UR - http://www.scopus.com/inward/record.url?scp=18244401901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18244401901&partnerID=8YFLogxK

U2 - 10.1007/s00605-004-0278-8

DO - 10.1007/s00605-004-0278-8

M3 - Article

AN - SCOPUS:18244401901

VL - 145

SP - 19

EP - 33

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 1

ER -