In 1981, F. Beukers proved the following theorem in his article On the generalized Ramanujan-Nagell equation I :
Theorem $\bf1$. $~$ Suppose $m \in \mathbb{Z}$, then for any integer $x$, $$ \left | \frac{x}{2^m} - \sqrt{2} \right | > 2^{-1.8m-43.9}. $$
By using Theorem $\bf 1$, he obtained:
Theorem $\bf 2$. $~$ Suppose $D \neq 0$ and $D \in \mathbb{Z}$. If the diophantine equation $$ x^2 - D = 2^n $$ has positive integer solution $(x,n)$, then $n < 435 + \frac{10 \log |D|}{\log 2}$.
Two questions come to my mind:
Firstly, is there an optimization of the upper bound of $n$ in Theorem $\bf 2$ ?
Secondly, if we make a small change in the above equation, namely, $x^3 - D = 2^n$, it may require inequalities like these to solve: $$ \left | \frac{x}{2^m} - \sqrt[3]{2} \right | > C_1(m), \tag{1} $$ and $$ \left | \frac{x}{2^m} - (\sqrt[3]{2})^2 \right | > C_2(m), \tag{2} $$ where $C_1(m)$ and $C_2(m)$ are some constants like in Theorem $\bf 1$.
Hence, my second question is: Are there some references on inequalities $(1)$ and $(2)$ ? If so, what are the best known lower bounds?
Supplements:
From his article Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers, A. Baker, 1964, I have found something useful to part of my questions but not all. In his introduction, he states a theorem of Roth that
''Suppose that $\alpha$ is an algebraic number, not rational, and that $\kappa > 2$. The well-known theorem of Roth implies the existence of $$ c=c(\alpha,\kappa)>0 $$ such that $$ \left | \alpha - \frac{p}{q} \right | > \frac{c}{q^{\kappa}} $$ for all rationals $\frac{p}{q} (q>0)$, but the method of proof does not allow $c$ to be determined explicitly...''
Hence, this result maybe a good choice for the proof of the finite solution of a large kinds of diophantine equations. But it is not an effective method for determining a computable range. Thanks to the work of A. Baker, he gives a corollary which says that
''For all rationals $\frac{p}{q} (q>0)$ we have $$ \left | \sqrt[3]{2} - \frac{p}{q} \right | > \frac{c}{q^{2.955}} , \tag{3} $$ where $c = 10^{-6}$.''
This can be directly applied to inequality $(1)$. However, any references with respect to inequality $(2)$ is still unknown for me, let alone other further references. This above is all what I have known now.
Edit $1$:
It is a nice surprise when I found this article: Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers revisited, P. M. Voutier, 2007. From it, I know a much better version of $(3)$ which was given by M. Bennett in 1997 is as follows: $$ \left | \sqrt[3]{2} - \frac{p}{q} \right | > \frac{0.25}{q^{2.45}}. $$ However, P. M. Voutier has obtained a wide range of such results ( Corollary $2.2$ ):
''For the values of $n$ given in Table $1$, we have $$ \left | \sqrt[3]{n} - \frac{p}{q} \right | > \frac{c_2}{|q|^{\kappa +1}}, $$ for all integers $p$ and $q$ with $q \neq 0$ where $c_2$ and $\kappa$ are the values corresponding to $n$ in Table $1$.''
The Table $1$ illustrates the values of $n$ from $2$ to $100$ ( not all continuous 99 numbers ) with corresponding constants $c_2$ and $\kappa$. As an example, if $n=4$, then $c_2 = 0.41$ and $\kappa = 1.4325$. This is a concrete example of $(2)$ if we take $p = x$ and $q = 2^m$.
The upper bound in your Theorem 1 has been sharpened to $$ n < 5.55 \log |D|, $$ provided $(n,D) \neq (3,-1)$ or $(15,7)$, in a paper of Bauer et al "Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation" (Ramanujan J 2002). Similar arguments allow one to prove reasonably good inequalities of the shape (1) or (2) with $C_1(m)$ or $C_2(m)$ roughly $2^{-1.61m}$. As far as I know, nothing along these lines has been published.