Outlined automata methods for generating Catalan numbers and supplied sample proofs.
Gathered open problems for future exploration without new computations.
Surveyed bounded prime gap techniques following arXiv:1408.5157.
Computed the chromatic polynomial of $C_5$ at $k=3$ giving $30$ colourings.
Hardy--Ramanujan's approximation $1.99\times10^8$ for $p(100)$ differed by $4.6\%$ from the exact $190{,}569{,}292$.
Confirmed the periodic continued fraction of $\sqrt{3}$ as $[1;\overline{1,2}]$.
Evaluated the Catalan number $C_5=42$ and verified the recurrence $C_{n+1}=\sum_{k=0}^{n}C_kC_{n-k}$ at $n=4$.
A hyperbolic triangle with angles $(\pi/3,\pi/4,\pi/6)$ encloses area $\pi/6$.
The reciprocal sum of primes up to $100$ reached $\sum_{p\le100}1/p\approx2.034$, matching $\ln\ln100+0.261$.
Prime counts $\pi(x)$ up to $100$ confirmed $\pi(100)=25$.
Euler characteristics $\chi=2-2g$ verified $\chi=0$ for $g=1$.
The first six Catalan numbers, stored in catalan.csv, included $C_4=14$.