On the diophantine equation $px^2+3^n=y^p$

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Fadwa S. Abu Muriefah

Abstract

Let $p$ be a prime. In this paper we prove that: (1) the equation $px^2+3^{2m}=y^p$, $p\not\equiv 7$ (mod 8) has no solutions in positive integers $(x,y,p)$ with $\hbox{gcd}(3,y)=1$. (2) if $3|y$ then the equation has at most one solution when $p=3$ and for $p>3$ it may have a solution only when $p\equiv 2$ (mod 3) and $\hbox{gcd}(p,m)=1$. (3) the equation $px^2+3^{2m+1}=y^p$, $p\not\equiv 5$ (mod 8), has a solution only when $p=3$, and this solution exists only when $m=3+3M$, and is given by $x=46.3^{3M+1}$, $y=13.3^{2M+1}$.

Article Details

How to Cite
Muriefah, F. S. A. (2000). On the diophantine equation $px^2+3^n=y^p$. Tamkang Journal of Mathematics, 31(1), 79–84. https://doi.org/10.5556/j.tkjm.31.2000.418
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Papers
Author Biography

Fadwa S. Abu Muriefah

Mathematics Department, Girls College of Education, P. O. Box 60561, Riyadh-11555, Saudi Arabia.