On a class of Lebesgue-Ljunggren-Nagell type equations

Abstract

Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $a{x}^2+b^{2l}=4y^n$, $x,y\in \mathbb{Z}$, $l\in \mathbb{N}$, $n$ odd prime, $\gcd (x,y)=1$. We completely solve the above Diophantine equation for $a\in \{7,11,19,43,67,163\}$, and $b$ a power of an odd prime, under the conditions $2^{n-1}{b}^l\not\equiv \pm 1(\mathrm{mod}a)$ and $\gcd (n,b)=1$. For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers $x$, $y$ with ($\gcd (x,y)=1$), $l\in \mathbb{N}$ and all odd primes $n>3$, satisfying $2^{n-1}{b}^l\not\equiv \pm 1(\mathrm{mod}a)$, $\gcd (n,b)=1$, and $\gcd (n,h(-a))=1$, where $h(-a)$ denotes the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-a})$.

accepted for publication in Journal of Number Theory