Parvathi S Nair, Sudhansu Sekhar Rout
Published 2024-11-17Version 1
Let $S$ be a fixed set of primes and let $(X_{l})_{l\geq 1}$ be the $X$-coordinates of the positive integer solutions $(X, Y)$ of the Pell equation $X^2-dY^2 = 1$ corresponding to a non-square integer $d>1$. We show that there are only a finite number of non-square integers $d>1$ such that there are at least two different elements of the sequence $(X_{l})_{l\geq 1}$ that can be represented as a sum of $S$-units with a fixed number of terms. Furthermore, we solve explicitly a particular case in which two of the $X$-coordinates are product of power of two and power of three.
Positive Integer Solutions of the Pell Equation $x^{2}-dy^{2}=N,$ $% d\in \left\{k^{2}\pm 4,\text{}k^{2}\pm 1\right\} $ and $N\in \left\{\pm 1,\pm 4\right\}
On $X$-coordinates of Pell equations which are repdigits
Solutions of the Pell equations x^2-(a^2+2a)y^2=N via generalized Fibonacci and Lucas numbers
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