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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 equation x+y = α in multiplicatively dependent unknowns

P. Habegger (2005)

Acta Arithmetica

The equation x+y=1 in finitely generated groups

F. Beukers, H. P. Schlickewei (1996)

Acta Arithmetica

The equation xyz = x + y + z = 1 in integers of a cubic field.

Andrew Bremner (1989)

Manuscripta mathematica

The equation xyz=x+y+z=1 in integers of a quartic field

Andrew Bremner (1991)

Acta Arithmetica

The Equations Defining Abelian Varieties and Modular Functions.

Shoji Koizumi (1979)

Mathematische Annalen

The Equations for Modular Function Fields of Principal congruence subgroups of prime level.

Nobuhiko Ishida, Noburo Ishii (1996)

Manuscripta mathematica

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

Let $f = \sum_{n = 1}^{\infty} a (n) q^{n} \in S_{k + 1 / 2} (N, χ_{0})$ be a nonzero cuspidal Hecke eigenform of weight $k + \frac{1}{2}$ and the trivial nebentypus $χ_{0}$ , where the Fourier coefficients $a (n)$ are real. Bruinier and Kohnen conjectured that the signs of $a (n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${a (t n^{2})}_{n}$ , where $t$ is a squarefree integer such that $a (t) \neq 0$ . Let $q$ and $d$ be natural numbers such that $(d, q) = 1$ . In this work, we show that ${a (t n^{2})}_{n}$ is equidistributed over any arithmetic progression $n \equiv d mod q$ .

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

The equation ${(j + k + 1)}^{2} - 4 k = Q n^{2}$ and related dispersions.

The equation n₁n₂ = n₃n₄, the gcd-sum function and the mean values of certain character sums

The equation n(n+d) ⋯ (n+(k-1)d) = by² with ω(d) ≤ 6 or d ≤ 10¹⁰

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

The equation x+y = α in multiplicatively dependent unknowns

The equation x+y=1 in finitely generated groups

The equation xyz = x + y + z = 1 in integers of a cubic field.

The equation xyz=x+y+z=1 in integers of a quartic field

The Equations Defining Abelian Varieties and Modular Functions.

The Equations for Modular Function Fields of Principal congruence subgroups of prime level.

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

The Erdhős-Kac Theorem for Curvatures in Integral Apollonian Circle Packings

The Erdős and Halberstam theorems for Drinfeld modules of any rank (with an appendix by Hugh Thomas)

The Erdős Theorem and the Halberstam Theorem in function fields

The Erdős-Ginzburg-Ziv theorem in Abelian non-cyclic groups.

The Erdős-Heilbronn problem for finite groups

The Erdős-Turán property for a class of bases

The ergodic theory of shrinking targets.

The Euclidean algorithm for Galois extensions of Q.

Currently displaying 361 – 380 of 1341