Cubic forms, powers of primes and the Kraus method

Volume 128 / 2012

Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów Colloquium Mathematicum 128 (2012), 35-48 MSC: 11D41, 11F80, 11G05. DOI: 10.4064/cm128-1-5

Abstract

We consider the Diophantine equation $(x+y)(x^2+Bxy+y^2)=Dz^p$, where $B$, $D$ are integers ($B\not =\pm 2$, $D\not =0$) and $p$ is a prime $>5$. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many $B$ and $D$) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with $B=0, 1, 3, 4, 5, 6$, specific $D$'s, and $p\in (10,10^{6})$). In the last section we discuss reductions of the above Diophantine equations to those of signature $(p,p,2)$.

Authors

  • Andrzej DąbrowskiInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail
  • Tomasz JędrzejakInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail
  • Karolina KrawciówInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail

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