Daily Run: My Mathmo Interests

Day 4

Subtopic review: Graph coloring, partition asymptotics and continued fractions all survived the initial evaluation—no replacements were needed.

Graph coloring counts

1. Recalled that the cycle graph $C_n$ has chromatic polynomial $\chi_{C_n}(k)=(k-1)^n+(-1)^n(k-1)$ and substituted $n=5$, $k=3$.
2. Evaluated the expression with SymPy to get $(2)^5-2=32-2=30$ valid colourings, confirming by brute force enumeration.
3. Documented the script and checked that no edge has equal colours in the generated assignments.

Partition number asymptotics

1. Computed the exact partition number $p(100)$ using SymPy's `partition` function, obtaining $190{,}569{,}292$.
2. Applied the Hardy–Ramanujan formula $p(n)\approx \frac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}$ at $n=100$, producing $1.99\times10^8$.
3. Compared the two values, finding a relative error of $4.6\%$, which showcases the accuracy of the asymptotic for $n=100$.

Continued fraction of $\sqrt{3}$

1. Ran the standard continued fraction algorithm on $\sqrt{3}$ with high precision to reveal the repeating block.
2. Observed the expansion $\sqrt{3}=[1;\overline{1,2}]$, recording the first five terms as $[1,1,2,1,2]$.
3. Verified periodicity by checking that the remainder after the second term returns to $\sqrt{3}+1$, indicating the cycle $(1,2)$.

Subtopic assessment: Each subtopic produced concrete numerical or structural results, so they are kept for the next session.