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)

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 .