Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski¹ ; Tomasz Jędrzejak¹ ; Karolina Krawciów¹

¹ Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland

Colloquium Mathematicum, Tome 128 (2012) no. 1, pp. 35-48.

Résumé

We consider the Diophantine equation $(x+y)(x^2+Bxy+y^2)=D{z}^p$, where $B$, $D$ are integers ($B\ne \pm 2$, $D\ne 0$) 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 $(p,p,2)$.

Détail

DOI : 10.4064/cm128-1-5

Mots-clés : consider diophantine equation bxy where integers prime kraus type criteria nonsolvability equation explicitly many terms galois representations modular forms apply these criteria numerous equations specific section discuss reductions above diophantine equations those signature