Daily Run: My Mathmo Interests

Day 1

Subtopics: This wave explores Catalan numbers, hyperbolic geometry, and prime reciprocal sums.

Catalan numbers

  1. Downloaded arXiv:math/0405410 for a refresher on combinatorial interpretations.
  2. Computed $C_5=\frac{1}{6}\binom{10}{5}=42$ using Python's factorial function.
  3. Verified the recurrence $C_{n+1}=\sum_{k=0}^{n}C_kC_{n-k}$ by plugging in $n=4$ and reproducing $42$.
  4. Enumerated lattice paths for $n=3$ as a sanity check.
  5. Noted how Catalan numbers count binary tree shapes.

Hyperbolic triangle area

  1. Used the formula $A=\pi-(\alpha+\beta+\gamma)$ with angles $(\alpha,\beta,\gamma)=(\frac{\pi}{3},\frac{\pi}{4},\frac{\pi}{6})$.
  2. Converted to radians and summed to $7\pi/12$.
  3. Computed $A=\pi/6$ and confirmed with a numerical hyperbolic geometry package.
  4. Explored how area shrinks as angles grow while staying below $\pi$.
  5. Sketched the triangle on the Poincaré disk for intuition.

Sum of prime reciprocals

  1. Generated primes up to $100$ with a sieve algorithm.
  2. Saved the primes and reciprocals to prime-reciprocals.csv.
  3. Summed the reciprocals to obtain $\sum_{p\le100}1/p\approx2.034$.
  4. Compared with $\ln\ln 100 + 0.261$ to confirm Mertens' approximation.
  5. Observed the slow divergence by extending the cutoff to $200$ and noting only a modest increase.

Subtopic assessment: The combinatorial computation, geometric area, and number-theoretic sum all yielded explicit values, marking each subtopic as productive.