For myself I would like to bring up Leonard Lewin's Polylogarithms and associated function. In chapter $2$, dealing with the Inverse Tangent Integral, Catalan's Constant is introduced:
The value of $\operatorname{Ti}_2(1)$ cannot, however, be deduced from this functional relation, and so far as is known is a new constant of analysis, denoted by $G$ known as Catalan's constant. It occurs in various contexts, and its value, to $8$ decimal places, is
$$\operatorname{Ti}_2(1)~=~G~=~0.91596559\tag{2.7}$$
Fascinating about this constant I would claim is especially its broad occurrence within many, on the first sight distinct, problems of closed-form integration aswell as closed-form summation. As José Carlos Santos already pointed out: whilst taking into account how less we know about this real number it is odd how widely it can be used.
I firstly encoutered this constant in the context of Dilogarithms, to be precise by invoking auxiliary functions such as the aforementioned Inverse Integral Tangent or the Clausen Functions, and the Dirichlet Beta Function which both tend to be possible defintions for the constant in terms of an infinite series, namely
$$G~=~\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2}\tag1$$
Even though this sum looks pretty similiar to the Riemann Zeta Function and its relatives it cannot be related to them that simple $($the only possible functional relation I am aware of is given within this article but hence it uses the Polygamma Functions aswell I would not call this relation "easy" as e.g. the one between the Riemann Zeta Function and the Dirichlet Eta Function$)$. To speak for myself the different "character" of the Dirichlet Beta Function, i.e. having expressible values for odd positive integers, not being expressable with the help of the Riemann Zeta Function alone, etc., justifies its importance.
Of course, only to appear everywhere is not a criterion for being important alone but appearing over and over again, especially in connection with $\pi$, seems to imply that there is something more about this number. Concerning the appearance I would like to refer to this question here on MSE dealing with the relationship between Catalan's Constant and $\pi$ in particular.
To bring up another point showing the importance of Catalan's Constant I would claim that its role can be compared with the role of Apéry's Constant $\zeta(3)$. Both can be defined in terms of infinite series, namely the Dirichlet Beta Function and the Riemann Zeta Function respectively. Both are the first positive integers for which the underlying Function cannot be expressed using other constants. As already mentioned there are formulae for the even positive integer values of the Riemann Zeta Function and for the odd positive integer values of the Dirichlet Beta Function given by
\begin{align*}
\zeta(2n)~&=~(-1)^{n-1}\frac{(2\pi)^n}{2(2n)!}\operatorname{B}_{2n}\tag2\\
\beta(2n+1)~&=~(-1)^n\frac{\pi^{2n+1}}{4^{n+1}(2n)!}\operatorname{E}_{2n}\tag3
\end{align*}
Whereas we defined $\beta(2)$ as Catalan's Constant and $\zeta(3)$ as Apéry's Constant. A crucial different, however, is that we know that the latter constant is in fact irrational due to Apéry who proved this back in $1979$. Interesting enough a similiar series representation for $G$ exists like the one Apéry used within his proof. These series are given by
\begin{align*}
\zeta(3)~&=~\frac52\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}n}\tag4\\
\beta(2)~&=~\frac\pi8\ln(2+\sqrt 3)+\frac38\sum_{n=0}^\infty \frac1{(2n+1)^2\binom{2n}n}\tag5
\end{align*}
All in all Catalan's Constant does not only occurs in a bunch of mathematical problems related to integration and summation but also plays a quite important role in the field of Zeta Functions and relatives. I would say especially the striking parallels between Apéry's Constant and Catalan's Constant consolidate the importance of this constant.
Best Answer
$a^n-1$ is always divisible by $a-1$, and hence non-prime, or composite $\ldots$ Oh, wait ! Unless $a-1$ $=1\iff a=2$. $($This explains the mathematical interest in Mersenne primes$)$. Also, $a^n+1$ is always divisible by $a+1$, and hence non-prime, or composite $\ldots$ unless $n=2^k$. $($This explains the mathematical interest in Fermat primes; see also$)$.
See my answer to this question.