Cubic forms, powers of primes and the Kraus method
Colloquium Mathematicum, Tome 128 (2012) no. 1, pp. 35-48
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
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 $(p,p,2)$.
Keywords:
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
Affiliations des auteurs :
Andrzej Dąbrowski 1 ; Tomasz Jędrzejak 1 ; Karolina Krawciów 1
@article{10_4064_cm128_1_5,
author = {Andrzej D\k{a}browski and Tomasz J\k{e}drzejak and Karolina Krawci\'ow},
title = {Cubic forms, powers of primes and the {Kraus} method},
journal = {Colloquium Mathematicum},
pages = {35--48},
year = {2012},
volume = {128},
number = {1},
doi = {10.4064/cm128-1-5},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm128-1-5/}
}
TY - JOUR AU - Andrzej Dąbrowski AU - Tomasz Jędrzejak AU - Karolina Krawciów TI - Cubic forms, powers of primes and the Kraus method JO - Colloquium Mathematicum PY - 2012 SP - 35 EP - 48 VL - 128 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4064/cm128-1-5/ DO - 10.4064/cm128-1-5 LA - en ID - 10_4064_cm128_1_5 ER -
Andrzej Dąbrowski; Tomasz Jędrzejak; Karolina Krawciów. Cubic forms, powers of primes and the Kraus method. Colloquium Mathematicum, Tome 128 (2012) no. 1, pp. 35-48. doi: 10.4064/cm128-1-5
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