In 1976 Tijdeman proved that the Catalan equation
$$
x^{p}-y^{q}=1
$$
has finitely many solutions in integers $x,y,p,q>1$ in his paper


* R. Tijdeman, *On the equation of Catalan*, Acta Arith. **29** (1976) pp 197–209 ([EuDML](https://eudml.org/doc/205418))


He just found the following upper bound for $p$ and $q$ using Baker theorem in linear form in logarithm


\begin{align}
p& <2c\_{9}(\log p)^{c\_{10}}\\
q& <c\_{1}(\log p)^{c\_{2}}
\end{align}


I don't understand how these two inequalities for $p$ and $q$ give us that Catalan's equation has only a finite number of solution since he didn't give an upper bound for $x,y$.


Also in 1993, Overholt showed that Brocard equation
$$
n!+1=m^{2}
$$
has finitely solution if Szpiro's conjecture is true.
He just found that $n<4^{\epsilon}e $.
I don't understand how this upper bound for $n$ make us say that the Brocard equation has finitely many solutions?


I ask if a finding of an upper bound for a
least one variable of an arbitrary Diophantine equation is enough to prove that it has only finitely many solutions in $\mathbb{Z}$? If yes does the upper should do not depend on the other variable of that Diophantine equation?


*Edit* I need to answer the third question just .



How to prove that a diophantine equation has only finitely many solutions in integers?
By the initial remarks in the paper, one can restrict to $p,q\geq 5$. By Theorem A in the paper (which is a result of Baker's from 1969), $x$ and $y$ can be effectively bounded in terms of $p$ and $q$:
$$\max(|x|,|y|)<\exp\exp(5^{10}p^{10} q^{10q^3}).$$


Hence it suffices to bound $p$ and $q$.


