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On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power
- Source :
- Functiones et Approximatio Commentarii Mathematici. 64
- Publication Year :
- 2021
- Publisher :
- Adam Mickiewicz University (Euclid), 2021.
-
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 on 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 $f(x)\in\mathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.<br />Comment: We have completely changed the methodology compare to the first version. For fix $d$ the new approach has already been used by Vandita Patel and the author in an earlier paper https://doi.org/10.1142/S1793042118501646 and we extend it when $d$ is an arbitrary prime power
Details
- ISSN :
- 02086573 and 17930421
- Volume :
- 64
- Database :
- OpenAIRE
- Journal :
- Functiones et Approximatio Commentarii Mathematici
- Accession number :
- edsair.doi.dedup.....9b6cdec0d2425ed8de20dac2fe1880d5