Address:
Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India
Department of Mathematics, Faculty of Science, Rangapara College, Rangapara, Sonitpur-784505, Assam, India
E-mail: baruah067@gmail.com anup@gauhati.ac.in>
Abstract: We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\ge 0, n \ge 3$ and $x, y\ge 1$ are unknown integers with $\gcd (x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.
AMSclassification: primary 11D61; secondary 11D41, 11Y50.
Keywords: Diophantine equation, Lehmer sequence, elliptic curve, quartic curve, S-integral points.
DOI: 10.5817/AM2024-3-135