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On generalized Fermat equations of signature (p,p,3)

Karolina Krawciów — 2011

Colloquium Mathematicae

This paper focuses on the Diophantine equation $x ⁿ + p^{α} y ⁿ = M z ³$ , with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n \geq ℱ (M, p^{α})$ , where $ℱ (M, p^{α})$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].

Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski; Tomasz Jędrzejak; Karolina Krawciów — 2012

Colloquium Mathematicae

We consider the Diophantine equation $(x + y) (x ² + B x y + y ²) = D z^{p}$ , 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).

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