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June 2021 On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power
Angelos Koutsianas
Funct. Approx. Comment. Math. 64(2): 141-151 (June 2021). DOI: 10.7169/facm/1805

Abstract

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors of Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb{Z}[x]$, $a$ is a positive integer and $b$ an arbitrary positive integer.

Citation

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Angelos Koutsianas. "On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power." Funct. Approx. Comment. Math. 64 (2) 141 - 151, June 2021. https://doi.org/10.7169/facm/1805

Information

Published: June 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.7169/facm/1805

Subjects:
Primary: 11D61

Keywords: exponential equation , Lehmer sequences , primitive divisors

Rights: Copyright © 2021 Adam Mickiewicz University

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Vol.64 • No. 2 • June 2021
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