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Keywords:
Diophantine equation; Lehmer sequence; elliptic curve; quartic curve; S-integral points
Diophantine equation; Lehmer sequence; elliptic curve; quartic curve; S-integral points
Summary:
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.
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