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

K. Györy, L. Hajdu, N. Saradha

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 languageEnglish
Pages (from-to)373-388
Number of pages16
JournalCanadian Mathematical Bulletin
Volume47
Issue number3
Publication statusPublished - Sep 2004

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Diophantine equation
Arithmetic sequence
Term
Common difference
Integral Solution
Rational Solutions
Coprime
Euler
Consecutive
Theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the diophantine equation n(n + d)... (n + (k - 1)d) = byl. / Györy, K.; Hajdu, L.; Saradha, N.

In: Canadian Mathematical Bulletin, Vol. 47, No. 3, 09.2004, p. 373-388.

Research output: Contribution to journalArticle

Györy, K. ; Hajdu, L. ; Saradha, N. / On the diophantine equation n(n + d)... (n + (k - 1)d) = byl. In: Canadian Mathematical Bulletin. 2004 ; Vol. 47, No. 3. pp. 373-388.
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