Let $A=\{a_i\in\mathbb{Q}:a_i> 1\land 1\le i \le b-2\}$, (cardinality of $A$ is $b-2$) and $\displaystyle B=\bigcup_{c=0}^{\infty} \left[\dfrac{bc+1}{d},\dfrac{b(c+1)-1}{d}\righ …

Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares of le …

Terence Tao asked for a non-enumerative proof that a positive proportion of permutations are derangements and got a great answer. Inspired by this, I'd like to ask about another fa …

Denote the number of derangements by $D_N$. It's known that $D_N/N! \rightarrow 1/e$. Therefore $N!/e$ is an approximation for $D_N$. I'm trying to bound the difference between th …

We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p-1$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle …

It is easy to establish that $$ \left({n\choose k}\right)=(-1)^k{-n \choose k}, $$ where the symbol on the left-hand-side counts the number of multisets of $k$ elements from $n$. …

A row of $\mathbb{L}$ empty cells is available to represent a number as follows Number 1 : -------- Choose any cell at random and add 1. 2 …

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highe …

What is the tightest upper bound we can establish on the central binomial coefficients $ 2n \choose n$ ? I just tried to proceed a bit, like this: $ n! > n^{\frac{n}{2}} $ for a …

For a fixed finite alphabet $A=\{a,b,...\}$, write $x \sim_n y$ if the two words $x$ and $y$ have the same (scattered) subwords of length at most $n$. The relation $\sim_n$ is a co …

Advanced sudoku-solving seems rather streamlined: for each square write down the acceptable values Using a standard set of techniques - many of which I did not know by name - ded …

What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). …

I have $k$ distinct prime numbers $\ell_1 < \dots <\ell_k$, and for each $i=1,\dots,k$, a subset $A_i$ of $\mathbb Z / \ell_i \mathbb Z$. Let $L=\ell_1 \dots \ell_k$. Now usi …

The inequality is \begin{equation*} \sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right) \end{equation*} for all integer $d\geq 1$. I use com …

Are there any results known about the discriminants of indefinite integral binary quadratic forms admitting automorphisms of order 3 or 6? It seems reasonable to expect that any p …