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

100%

Krawciów K.

Colloquium Mathematicum

2011

tom 123

nr 1

49-52

This paper focuses on the Diophantine equation $xⁿ+p^{α}yⁿ = Mz³$, with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n ≥ ℱ(M,p^{α})$, 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

51%

Dąbrowski A., Jędrzejak T., Krawciów K.

Colloquium Mathematicum

2012

tom 128

nr 1

35-48

We consider the Diophantine equation $(x+y)(x²+Bxy+y²) = Dz^{p}$, 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).

Ograniczanie wyników

2 Colloquium Mathematicum

2 Krawciów K.

1 Dąbrowski A.

1 Jędrzejak T.

1 2012

1 2011

Wyniki wyszukiwania

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

Cubic forms, powers of primes and the Kraus method