Perfect powers from products of consecutive terms in arithmetic progression

K. Györy, L. Hajdu, Á Pintér

Research output: Contribution to journalArticle

20 Citations (Scopus)

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 languageEnglish
Pages (from-to)845-864
Number of pages20
JournalCompositio Mathematica
Volume145
Issue number4
DOIs
Publication statusPublished - Jul 2009

Fingerprint

Arithmetic sequence
Ternary
Consecutive
System of equations
Term
Galois Representations
Cyclotomic
Sieve
Solvability
Curve
Integer

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

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

In: Compositio Mathematica, Vol. 145, No. 4, 07.2009, p. 845-864.

Research output: Contribution to journalArticle

Györy, K. ; Hajdu, L. ; Pintér, Á. / Perfect powers from products of consecutive terms in arithmetic progression. In: Compositio Mathematica. 2009 ; Vol. 145, No. 4. pp. 845-864.
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