Author Archives: Brian

One of my favorite facts from combinatorics is that $\sum\limits_{0\leq k\leq n}\binom{n}{k} = 2^{n}$. To prove it is simple: Note that $2 = 1 + 1$, and that $\binom{n}{k} = \binom{n}{k}\cdot 1^{k}\cdot 1^{n-k}$, and appeal to binomial theorem: $$\sum_{0\leq k\leq … Continue reading →

For each real number $t>1$, one can define a probability distribution $P_{t} = \left\{\mathrm{pr}_{t}\left(n\right)\right\}_{n\in\mathbb{N}}$ by $\mathrm{pr}_{t}\left(n\right) = \displaystyle{\frac{n^{-t}}{\zeta\left(t\right)}}$. This class of probability functions was studied by Golomb in [1]. In this post, we will prove a theorem about relatively $r$-prime … Continue reading →

I was recently introduced to an interesting polynomial problem. Problem. Determine all polynomials $p\left(x\right)\in \mathbb{R}\left[x\right]$ such that $\left(x – 16\right)p\left(2x\right) = 16\left(x – 1\right)p\left(x\right)$ for all $x\in \mathbb{R}$.

While reading some course notes from MIT 18.703 (Modern Algebra), I came across the following statement on page 3: Lemma 22.3. The product of all primes $r$ between $N$ and $2N$ is greater than $2^{N}$. However, this can quickly be … Continue reading →

In two papers of Nymann and Leahey from the mid-to-late 70’s, they determined the asymptotics of the probabilities of selecting $k$ relatively prime integers [1] and selecting an integer which is $k$-free [2] according to the uniform and binomial distributions. … Continue reading →

Yesterday, I was made aware of an elementary combinatorics problem, and the solution was surprisingly clever (at least it is to me, though I am not an expert in the area). Suppose that a philanthropist decides to give a total … Continue reading →

I saw this post on Reddit and was quite interested in it. I decided to investigate things on my own for a bit. We’ll start with the statement of the Cauchy Condensation Test.