## Posts

### Welcome

Hi, Welcome to my little math blog. This is where I will dump thoughts on recreational mathematics, fun puzzles, and other stuff every so often. Thanks for stopping by! Hopefully you find something to enjoy here.
Recent posts

### A silly little derivation of $$\zeta(2)$$

(This is a cleaned-up and somewhat expanded version of this Twitter thread .) What follows is a silly little proof that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ where $$\zeta$$ is the Riemann zeta function. Consider the integral $I := \int_0^1 \frac{\log(1 - x + x^2)}{x(x - 1)} \, dx.$ We have, by using partial fractions and performing some other algebraic manipulations, \begin{align*} I &= -\int_0^1 \! \frac{\log(1 - x + x^2)}{x} \, dx - \int_0^1 \! \frac{\log(1 - x + x^2)}{1 - x} \, dx \\ &= -2\int_0^1 \! \frac{\log(1 - x + x^2)}{x} & (x \mapsto 1 - x ) \\ &= 2\left( \int_0^1 \! \frac{\log(1 + x)}{x} \, dx - \int_0^1 \! \frac{\log(1 + x^3)}{x} \, dx \right) \\ &= \frac{4}{3}\int_0^1 \frac{\log(1 + x)}{x} \, dx & (x \mapsto x^{1/3}). \end{align*} To evaluate this integral, we take the Maclaurin series: $\int_0^1 \! \frac{\log(1 + x)}{x} \, dx = \int_0^1 \! \sum_{n=1}^{\infty} \frac{(-1)^nx^{n-1}}{n} \, dx$ Since for

### On My Favorite Number, 76923 (A Brief Survey of Cyclic Numbers)

(This is a cleaned-up, somewhat revised/expanded version of my Twitter thread here .) Among math enthusiasts, the number $$142857$$ is pretty cool. Move its leftmost digit to the right, and you get $$428571$$, which is three times the original: $$428571 = 142857 \times 3$$. Do this again, and you get $$285714$$, which is two times the original: $$285714 = 142857 \times 2$$. We can keep doing this until we return to $$142857$$, as follows: \begin{align*} 142857 &= 142857 \times 1 & 142857 \times 1 &= \color{red} 142857 \\ 428571 &= 142857 \times 3 & 142857 \times 2 &= 2857\color{red}14 \\ 285714 &= 142857 \times 2 & 142857 \times 3 &= 42857\color{red}1 \\ 857142 &= 142857 \times 6 & 142857 \times 4 &= 57\color{red}1428 \\ 571428 &= 142857 \times 4 & 142857 \times 5 &= 7\color{red}14285 \\ 714285 &= 142857 \times 5 & 142857 \times 6 &= 857\color{red}142 \end{align*} Numbers that give you consecutive