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We show that the Diophantine equation of the title has, for $n > 1$ , no solution in coprime nonzero integers $x, y$ and $z$ . Our proof relies upon Frey curves and related results on the modularity of Galois representations.

The Fermat equation with polynomial values as base variables.

B. Brindza, K. Györy, R. Tijdeman (1985)

Inventiones mathematicae

The first case of Fermat's last theorem.

D.R. Heath-Brown, L.M. Adleman (1985)

Inventiones mathematicae

The lattice points of an n-dimensional tetrahedron.

Andrew Granville (1991)

Aequationes mathematicae

The lattice points of an n-dimensional tetrahedron. (Summary).

Andrew Granville (1991)

Aequationes mathematicae

The method of infinite ascent applied on $A^{4} \pm n B^{3} = C^{2}$

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

Each of the Diophantine equations $A^{4} \pm n B^{3} = C^{2}$ has an infinite number of integral solutions $(A, B, C)$ for any positive integer $n$ . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$ , $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A, B, C)$ of the Diophantine equation $a A^{3} + c B^{3} = C^{2}$ for any co-prime integer pair $(a, c)$ .

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The abc-inequality and the generalized Fermat equation in function fields

The arithmetic of Fermat curves.

The Diophantine equation f(x) = g(y)

The diophantine equation $n^{i} + 1 = k (d n - 1)$ .

The diophantine equation $x^{2} + 3^{m} = y^{n}$ .

The diophantine equation x² + C = yⁿ

The distribution of primes satisfying the condition $a^{p - 1} \equiv 1 (m o d p^{2})$

The equation $a x^{m} + b y^{m} = c x^{n} + d y^{n}$

The equation $x^{2 n} + y^{2 n} = z^{5}$

The Fermat equation with polynomial values as base variables.

The first case of Fermat's last theorem.

The lattice points of an n-dimensional tetrahedron.

The lattice points of an n-dimensional tetrahedron. (Summary).

The method of infinite ascent applied on $A^{4} \pm n B^{3} = C^{2}$

The Mordell conjecture revisited

The number of solutions of the Thue-Mahler equation.

The power of powerful numbers.

The Tarry-Escott and the "easier" Waring problems.

The Way to the Proof of Fermat’s Last Theorem

Théorème de Brun-Titchmarsh; application au théorème der Fermat.

Currently displaying 1 – 20 of 28