A computational approach for solving $y^2=1^k+2^k+\dotsb+x^k$

DOI10.1090/S0025-5718-03-01465-0zbMath1113.11075OpenAlexW2000495889MaRDI QIDQ4417184

Ákos Pintér, Peter Gareth Walsh, Michael J. Jacobson jun.

Publication date: 28 July 2003

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0025-5718-03-01465-0

zbMATH Keywords

elliptic curvesquadratic fieldsDiophantine equations

Mathematics Subject Classification ID

Computer solution of Diophantine equations (11Y50) Algebraic number theory computations (11Y40) Cubic and quartic Diophantine equations (11D25) Exponential Diophantine equations (11D61) Linear forms in logarithms; Baker's method (11J86)

Related Items (11)

On the Diophantine equation \((x+1)^2+(x+2)^2+\ldots+(x+d)^2=y^n\) ⋮ On the Diophantine equation \((x+1)^{k}+(x+2)^{k}+\ldots+(2x)^{k}=y^{n}\) ⋮ Polynomial values of (alternating) power sums ⋮ Unnamed Item ⋮ A fast, rigorous technique for computing the regulator of a real quadratic field ⋮ On the power values of power sums ⋮ Subexponential time relations in the class group of large degree number fields ⋮ On the equation \(1^{k}+2^{k}+\cdots +x^{k}=y^{n}\) for fixed \(x\) ⋮ On a conjecture of Schäffer concerning the equation \(1^k + \ldots + x^k = y^n\) ⋮ On the congruence \(1^m + 2^m + \ldots + m^m\equiv n \bmod m\) with \(n\mid m\) ⋮ Perfect powers in sum of three fifth powers

Uses Software

• LiDIA

Cites Work

• Modular arithmetic on elements of small norm in quadratic fields
• The equation \(1^p+2^p+3^p+\ldots+n^p=m^q\)
• Correction to: On the diophantine equation \(1^ k+2^ k+\dots +x^ k+R(x)=y^ z\)
• The Size of the Fundamental Solutions of Consecutive Pell Equations
• The Diophantine equation x⁴ - Dy² = 1, II
• Zeros of Bernoulli, generalized Bernoulli and Euler polynomials
• 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 $1^k + 2^k + ... + x^k = y^z$
• The Diophantine equation $b^2X^4-dY^2=1$
• Computing integral points on elliptic curves
• An Investigation of Bounds for the Regulator of Quadratic Fields
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