The equation $(x-d)^5+x^5+(x+d)^5=y^n$

DOI10.4064/aa200617-11-11zbMath1475.11049arXiv2006.10349OpenAlexW3133356334MaRDI QIDQ4988887

Angelos Koutsianas, Michael A. Bennett

Publication date: 20 May 2021

Published in: Acta Arithmetica (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2006.10349

zbMATH Keywords

modularity of elliptic curvesFrey-Hellegouarch curvesSchinzel-Tijdeman

Mathematics Subject Classification ID

Exponential Diophantine equations (11D61) Higher degree equations; Fermat's equation (11D41)

Related Items (2)

Unnamed Item ⋮ Perfect powers in sum of three fifth powers

Uses Software

• Magma
• three_fourth_powers_code

Cites Work

• On powers that are sums of consecutive like powers
• The equation \(1^p+2^p+3^p+\ldots+n^p=m^q\)
• On the power values of power sums
• On the Diophantine equation \(1^k+2^k+\dots +x^k+R(x)=y^2\)
• Addendum to a paper of Harada and Lang
• A note on the equation \(1^ k+2^ k+\cdots+(x-1)^ k=y^ m\)
• The Magma algebra system. I: The user language
• On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
• Lucas sequences whose 12th or 9th term is a square
• Modular elliptic curves and Fermat's Last Theorem
• Ring-theoretic properties of certain Hecke algebras
• Klein forms and the generalized superelliptic equation
• On perfect powers that are sums of cubes of a seven term arithmetic progression
• On a conjecture of Schäffer concerning the equation \(1^k + \ldots + x^k = y^n\)
• On perfect powers that are sums of cubes of a five term arithmetic progression
• On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises
• Superelliptic equations arising from sums of consecutive powers
• On the Diophantine equation (x-1)^k+x^k+(x+1)^k=y^n
• Two-cover descent on hyperelliptic curves
• The Mordell–Weil sieve: proving non-existence of rational points on curves
• Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation
• On perfect powers that are sums of cubes of a three term arithmetic progression
• A diophantine equation
• On the equation $f(1)1^k + f(2)2^k + ... + f(x)x^k + R(x)= by^z$
• On some generalizations of the diophantine equation $1^k + 2^k + ... + x^k = y^z$
• On the equation $y^m = P(x)$
• On the equation $1^k + 2^k + ... + x^k = y^z$
• Sur l'équation a³+ b³= c^p
• Chabauty methods using elliptic curves
• Ternary Diophantine Equations via Galois Representations and Modular Forms
• On the Diophantine equation (x − d)4 + x4 + (x + d)4 = yn
• On the Diophantine equation $1^k+2^k+\dotsb+x^k=y^n$
• Perfect powers that are sums of squares in a three term arithmetic progression
• On the sum of fourth powers in arithmetic progression
• On a Diophantine equation

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