We consider the Diophantine equation , where B, D are integers (B ≠ ±2, D ≠ 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 ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-1-5,
author = {Andrzej D\k abrowski and Tomasz J\k edrzejak and Karolina Krawci\'ow},
title = {Cubic forms, powers of primes and the Kraus method},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {35-48},
zbl = {1314.11013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-1-5}
}
Andrzej Dąbrowski; Tomasz Jędrzejak; Karolina Krawciów. Cubic forms, powers of primes and the Kraus method. Colloquium Mathematicae, Tome 126 (2012) pp. 35-48. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-1-5/