### Abstract

We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Gyory on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, i ≥ 2 are fixed and k + l > 6.

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

Pages (from-to) | 373-388 |

Number of pages | 16 |

Journal | Canadian Mathematical Bulletin |

Volume | 47 |

Issue number | 3 |

Publication status | Published - Sep 2004 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{l}.

*Canadian Mathematical Bulletin*,

*47*(3), 373-388.

**On the diophantine equation n(n + d)... (n + (k - 1)d) = by ^{l}.** / Györy, K.; Hajdu, L.; Saradha, N.

Research output: Contribution to journal › Article

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*Canadian Mathematical Bulletin*, vol. 47, no. 3, pp. 373-388.

^{l}. Canadian Mathematical Bulletin. 2004 Sep;47(3):373-388.

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TY - JOUR

T1 - On the diophantine equation n(n + d)... (n + (k - 1)d) = byl

AU - Györy, K.

AU - Hajdu, L.

AU - Saradha, N.

PY - 2004/9

Y1 - 2004/9

N2 - We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Gyory on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, i ≥ 2 are fixed and k + l > 6.

AB - We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Gyory on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, i ≥ 2 are fixed and k + l > 6.

UR - http://www.scopus.com/inward/record.url?scp=4043084948&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4043084948&partnerID=8YFLogxK

M3 - Article

VL - 47

SP - 373

EP - 388

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 3

ER -