Todo on PostDoc Problems

Combinatorial Nullstellensatz

Sun, 26 Nov 2023 12:27:01 +0000

I came across this fun theorem in a talk about tree colorings:

Combinatorial Nullstellensatz. Let $f \in F[x_1, x_2, \ldots, x_n]$ be a polynomial of degree $t_1 + \cdots + t_n$. If $S_1, S_2, \ldots, S_n$ are nonempty subsets of $F$ such that $\left| S_i \right| \geq t_i + 1$ for all $i$, then there exists $s_1 \in S_1$, $s_2 \in S_2$, $\ldots$, $s_n \in S_n$ for which $$ f(s_1, s_2, \ldots, s_n) \neq 0 $$ as long as the coefficient of $x_1^{t_1} x_2^{t_2} \cdots x_n^{t_n}$ is nonzero.

Logarithms and primes

Thu, 26 Oct 2023 09:01:26 +0000

I came across this video by NumberPhile and 3Blue1Brown about the density of primes that are 1 mod 4 vs the density of primes that are 3 mod 4. The main fact is quite cool but there was something that he said during in the middle of the video that I found more interesting.

Define $\chi : \mathbb{N} \to {-1, 0, 1}$ as $$ \chi(n) = \begin{cases} 1 & \text{ if } n \equiv 1 \mod 4 \ -1 & \text{ if } n \equiv 3 \mod 4 \ 0 & \text{ otherwise. } \end{cases} $$ The first interesting fact that I do not know how to prove: $$ \sum \limits_{n \in \mathbb{N}} \dfrac{\chi(n)}{n} = \dfrac{\pi}{4}. $$ Note that the series is only conditionally convergent so the order of summation matters.

Every Prime $\equiv 1 \mod 4$ Is a Sum of Two Squares

Sun, 10 Sep 2023 15:35:44 +0000

I came across this “one sentence” proof that every prime congruent to 1 mod 4 can be written as a sum of two squares:

https://doi.org/10.2307/2323918

The proof involves constructing the following cryptic involution $$ (x, y, z) \mapsto \begin{cases}(x+2 z, z, y-x-z) & \text { if } x<y-z \ (2 y-x, y, x-y+z) & \text { if } y-z<x<2 y \ (x-2 y, x-y+z, y) & \text { if } x>2 y\end{cases} $$ on the set $S=\left{(x, y, z) \in \mathbb{N}^{3}: x^{2}+4 y z=p\right}$ and showing that the involution has exactly one fixed point $(1, 1, k)$ if and only if $p$ is a prime of the form $4k + 1$.