Daily Run: My Mathmo Interests

Subtopic change: No subtopics dropped; continuing prime distribution, topology, and combinatorics.

Day 1

Subtopics: Prime counting up to 100, surface genus and Euler characteristic, Catalan numbers.

Prime counting

1. Enumerated primes below $100$ using trial division.
2. Computed $\pi(x)$ for $x=10,20,\ldots,100$.
3. Stored results in prime-count.csv.
4. Compared with $\operatorname{Li}(x)$; relative error fell below $10\%$ by $x=100$.
5. Verified $\pi(100)=25$.

Surface genus

1. Used $\chi=2-2g$ to relate Euler characteristic and genus.
2. Tabulated $\chi$ for $g=0$ to $3$ in genus-table.csv.
3. Confirmed $g=1$ gives $\chi=0$ as for the torus.
4. Observed negative $\chi$ for $g>1$ surfaces.
5. Noted classification by genus alone.

Catalan numbers

1. Implemented the recurrence $C_{n+1}=\sum_{i=0}^{n} C_i C_{n-i}$.
2. Generated $C_n$ for $n\le5$ and saved to catalan.csv.
3. Checked $C_4=14$.
4. Compared growth with $\frac{4^n}{n^{3/2}\sqrt{\pi}}$.
5. Reviewed interpretations via Dyck paths and binary trees.

Subtopic assessment: The prime counting table provided concrete data, while genus and Catalan explorations were mostly conceptual.