On the finiteness of solutions for certain Diophantine equations

We study Diophantine equations of the form \( f(n)=m^2 \pm a,\; n, m\in \mathbb {N}, \) $f(n)=m^2\pm a,n,m\in \mathbb{N},$ where \(a\in \mathbb {N}^*\) $a\in {\mathbb{N}}^{\ast }$ and \(f: \mathbb {N} \rightarrow \mathbb {N}\) $f:\mathbb{N}\to \mathbb{N}$ tends to \(+\infty . \) $+\infty .$ Necessary and sufficient conditions for the set of solutions to be finite are formulated in terms of asymptotic properties and the repartition of the digits of the fractional part of \(\sqrt{f(n)}\) $\sqrt{f(n)}$