The Diophantine Equations $x^{2}-k=2\cdot T_{n}(\frac{b^{2}\pm2}{2})$

Gheorghe Udrea

Abstract: It is the object of this note to demonstrate that the two equations of the title have only finitely many solutions in positive integers $x$ and $n$ for any given integers $b$ and $k$, $k\ne\pm2$. In these equations $(T_{n}(x))_{n\ge0}$ is the sequence of the Chebyshev polynomials of the first kind.

Classification (MSC2000): 11B37, 11B39, 11D25

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