Let $Q_N$ denote the cubical cell complex given by the poset of faces of the $N$-cube, and let $$ V_N=\{\pm 1\}^N $$ be its set of $0$-cells. Let $$ A_N:=V_N/\{\pm 1\} $$ be the set of antipodal pairs of $0$-cells. We have $$ |A_N|=2^{N-1} $$ For each antipodal pair $$ \alpha=\{v,-v\}\in A_N $$...

It seems that for some integer $N$, namely any $N\equiv4\pmod {16}$, then they can be expressed as sum of $4$th powers of $4$ positive rational numbers. For example: $$15236 =\left(\frac{1875}{251}\right)^4+\left(\frac{11767}{3263}\right)^4+...

I would like to propose a prime-based variant of the Collatz conjecture that I came up with. I am interested to know if this specific variation has been studied before, or if there are any known counterexamples to the behavior I observed. Definition of the Function Let $n > 1$ be a positive...

I have designed a new variant of the Collatz conjecture based on modulo 4 remainders, and I am looking for computational data or heuristic analysis regarding its convergence. Definition of the Mapping Let $n$ be a positive integer. We define the transition function $g(n)$ as follows based on $n...

The curve $y^2=P(x)$ over the field of complex numbers, where $P(x)$ is a polynomial of degree $4$ without repeating roots, can be transformed with a birational transformation into $Y^2=Q(X)$ with $Q(x)$ of degree $3$ without repeating roots. That is, an elliptic curve. However, if $P(x)$ does...

Consider the recurrence $$ x_{n+1}=x_{n-1}+2\log x_n, $$ with positive initial values chosen so that the sequence remains positive and increasing. Since this may be rewritten as $$ \frac{x_{n+1}-x_{n-1}}{2}=\log x_n, $$ it resembles the centred-difference discretisation of the differential...

Suppose $a(n)$ is the minimum integer $k$ such that $k^{11}$ can be expressed as the sum of $n$ distinct positive 11th powers. Q: Find $a(n)$ for $10\le n\le 19$. For example, $a(20)=119$ because $199^{11}$ is the sum of 20 terms of $11$th powers,...

I've been having fun trying to find new number systems that aren't in the OEIS. One such number system, is the "Snowball Numbers", which I will define below. Apologies if these have been explored before, I could not find them. While playing with these numbers, I found the Catalan numbers (?!)...

How can I show that every Projective variety in $\mathbb{P}^n$ is compact in the induced Euclidean topology? Should I consider, as customary, an arbitrary open cover in $\mathbb{P}^n$, and perhaps use the projection map $\pi:\mathbb{C}^{n+1}\setminus\{0\}\rightarrow \mathbb{P}^n$ which defines...

Let $p_k$ denote the $k$-th prime number, and let $\operatorname{rev}(x)$ denote the decimal digit-reversal of a positive integer $x$. Define the consecutive-prime geometric sums: $$A_n = \sum_{i=0}^{p_n} p_{n+1}^i = \frac{p_{n+1}^{p_n + 1} - 1}{p_{n+1} - 1}, \qquad B_n = \sum_{i=0}^{p_n}...

I have been investigating an empirical variant of the classical Josephus problem and would like to know whether it has been studied previously. Let $J(n,k)$ denote the survivor of the classical Josephus problem with population size $n$ and elimination interval $k$. For fixed $n$, instead of...

Let $G$ be a reductive group over a field $k$. What actually does it mean to say that $G$ is of type $A_n, B_n,\dots,G_2,{}^2A_n, {}^3D_4,...$? In case it helps, I know what the Dynkin diagrams of types $A_n, B_n,\dots,G_2$ are (but not those of types ${}^2A_n, {}^3D_4,...$). I also know how to...

This statement is given as application of earlier results on the page $22 $ of the Daniel Perrin's Algebraic Geometry textbook from which I am self studying. Here $k$ is a commutative field and $V$ is a affine algebraic set. Application $6.9$ The morphism $\phi: k \to V=V(Y^2-X^3)$ given by...

Consider the sequence defined by $a_1 = 1$ and the recurrence relation for $n \ge 2$: $$a_n = \sum_{k=1}^{n-1} a_{\gcd(n,k)}$$ Grouping the terms by their divisors $d = \gcd(n,k)$, the number of integers $k < n$ such that $\gcd(n,k) = d$ is given by $\varphi(n/d)$, where $\varphi$ is Euler's...

Background & Definition In lattice geometry, two points $A, B \in \mathbb{Z}^2$ are said to be visible to one another if the open line segment between them contains no other lattice points. Equivalently, if $A = (x_A, y_A)$ and $B = (x_B, y_B)$, they are visible if $\gcd(|x_A - x_B|, |y_A -...

I have been analyzing a recursive sequence based on the greatest common divisor that acts as a dynamic sieve for twin primes. It shares structural similarities with Rowland's prime-generating sequence but targets the difference of squares. For any integer $n \ge 2$, define the sequence...

The Fermat quintic threefold is given by the equation, $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5=0\qquad\qquad$$ $\hskip1.5in$ (Incidentally, this threefold is a Calabi-Yau manifold, a type of manifold important to string theory.) In the integers, there are only four primitive solutions known, two which...

Let $f\in S_{k+1/2}^+(4q)$ be a newform in Kohnen’s space for $q$ an odd, square-free integer. For simplicity, assume that $k$ is even. Let $$f(z) = \sum_{\substack{n\geq 1\\ n\equiv 0,1\mod 4}}a_f(n)e(nz)$$ denote the Fourier expansion of $f$ at the cusp $\infty$. Let $D>0$ be a fundamental...

I am confused about the construction of the Hecke operators. I am defining them on $\Gamma_{1}(N)$ as $$(T_{m}f)(z) = \sum_{\substack{a,d \ge 1 \\ ad = m}}\langle a\rangle\left[\Gamma_{1}(N)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\Gamma_{1}(N)\right]_{k}f(z),$$ where $\langle a \rangle$ is...

For $U \to X$ étale, define a presheaf by $\mathcal{O}_{X_{\text{ét}}}(U) = \Gamma(U,\mathcal{O}_{U})$. I want to show that this is a sheaf on $X_{\text{ét}}$. Clearly, the sheaf condition holds for all Zariski open coverings, so it is sufficient to show the sheaf condition holds for étale...