Background The triangular grid places integer $T(r-1)+c$ at row $r$, column $c$, where $T(n)=n(n+1)/2$. Transposing this grid, reading along SE diagonals of the triangular grid as columns, yields a new array whose column $d$ has values $$f_d(n) = T(n+d-2)+n = \frac{n^2+(2d-1)n+(d-1)(d-2)/2 + ......

About prime factorization of consecutive integers, we all can notice prime factors vary apparently without any logic. Some numbers like $82 = 2 \times 41$ have highly unequal factors (high variance among the factors), while others like $80 = 2^4 \times 5$ or $2310 = 2 \times 3 \times 5 \times 7...

I am preparing a presentation and would appreciate feedback on the following explanation connecting the multiplicative group with Lubin–Tate theory. Let $K$ be a field and consider elements $x,y\in K^\times$ near the identity $1$. Write $$ x=1+X,\qquad y=1+Y. $$ Then the group law on $K^\times$...

Given a multiplicative function $f$ with divisor bound $|f|\le \tau_k$, where $k$ is a nonnegative real number. We consider the Dirichlet series $$ F(s)=\sum_{n=1}^\infty \dfrac{f(n)}{n^s}. $$ Since $$ \sum_{n=1}^\infty \dfrac{\tau_k(s)}{n^s}=\zeta(s)^k, $$ $F(s)$ absolutely converges on the...

Consider affine families $$ Q(u)=2^t3^{16}u+B, $$ with $t\ge 3$ and $2^t\mid 3B-1$. Set $$ C_0=\frac{3B-1}{2^t}. $$ Then $$ 3Q(u)-1 =2^t3^{17}u+(3B-1) =2^t(3^{17}u+C_0). $$ Suppose we restrict to a subfamily where, after the fixed factor $2^t$, another $2^\lambda$ divides the remaining factor:...

$\require{AMScd}$I am in the process of understanding how to work with (quotient) stacks. In my case, it is usually helpful to get my hands dirty so I like to come up with examples where maybe using stacks can make an argument more transparent. I recalled an exercise that I did when preparing...

Consider functions from $R^n$ to a subset of $R^n$. So $f(x_1,x_2,...,x_n) = (y_1,y_2,...,y_n)$. where $x_i,y_i$ are all real. More specific consider $$f(x_1,x_2,...,x_n) = (Q_1(x_1,x_2,...,x_n),Q_2(x_1,x_2,...,x_n),...,Q_n(x_1,x_2,...,x_n))$$ where the $Q_i$ are all Quadratic forms. Even more...

Numbers that are not yet mapped to are marked prime and given their own "trees", but really they are distance $\infty$ from the other primes. Traditionally, in a connected graph, BFS forms one tree, but really this is a collection of overlapping trees. What we have on the number line is a...

$$3(a^3+b^3+c^3+a+b+c)+5(a^2+b^2+c^2)+2(a^2b+b^2c+ac^2)+4(a^2c+bc^2+ab^2+ab+bc+ac)=0$$ A table of primitives known to me $(a, b, c)$, $a \in{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}^+$ $$ \boxed{\begin{array} {|r|r|r|r|}\hline №(a,b,c)& a_n & b_n & c_n \\ \hline S_1 & 0 & 0 & 0 \\ \hline S_2 & 0...

Let $$ \mathcal S=(\mathcal I,\Gamma,\Pi) $$ be a seam marked seed built from four compact oriented $2$-dimensional complex orbifold sheets. Each sheet is assumed to be a football type orbifold: its coarse underlying Riemann surface is $$ |\mathcal O_i|\cong \mathbb P^1 $$ and it has two...

Let $A\in GL(r,\mathbb{Z})$ be a positive definite matrix, $Q(x)=\dfrac{1}{2} x^t A x$ be the quadratic form associate to $A$, $Q^*(x)=\dfrac{1}{2} x^t A^{-1} x$ be the adjoint form of $Q(x)$ . Given $(c,d)=1,m\in\mathbb{Z}^r,$ we define $$G_m\left(\dfrac{d}{c}\right)=\sum_{h\,\text{mod}\,c}...

I apply the iterated forward difference operator to the sequence of primes; from $$ (p_n) = (2, 3, 5, 7, 11 \ldots) $$ I get $$ (d^1_n) = (1, 2, 2, 4 \ldots) $$ $$ (d^2_n) = (1, 0, 2 \ldots) $$ $$ (d^3_n) = (-1, 2 \ldots) $$ $$ \ldots $$ For each sequence $(x_n)$, one can reconstruct the...

Suppose that $$ x=u^{-d},\qquad y=f(u), $$ where $$ d\in \mathbb Z_{>0},\qquad f(u)\in \mathbb C((u)), \qquad y=o(x). $$ Equivalently, $\operatorname{ord}_u f(u)>-d$. Let $ (P,Q)\in \operatorname{Aut}_{\mathbb C}\mathbb C[x,y]$ be a polynomial coordinate system, and write $$ X=P(x,y),\qquad...

I am a curious outsider to mathematics and recently started reading about the Riemann Hypothesis. I am aware that many outsiders mistakenly believe they have solved the Riemann Hypothesis. I am not making such a claim. I am only trying to understand whether this perspective already exists in the...

I was solving this CSES question called Sum of Divisors and one of the solutions in the USACO Guide hints at this statement: The minimum value of $j > i$ such that $\lfloor n/j \rfloor$ < $\lfloor n/i \rfloor$ is $j = \lfloor n/q \rfloor + 1$, where $q = \lfloor n/i \rfloor$ for integers $j, i...

I am currently working through Fulton's Algebraic Curves and I have attempted the following problem: $$\text{Let P be a simple (non-singular) point on }F\text{ . Show that the tangent line to }F \text{ at } P\text{ has the equation }F_X (P )X + F_Y (P )Y + F_Z (P )Z = 0$$ My solution thus far...

In a prior post, the equation, $$x_1^5+2x_2^5 = y_1^5+2y_2^5$$ was considered. It has only one known primitive solution. This present post considers the similar, $$x_1^k+x_2^k+x_3^k = y_1^k+2y_2^k$$ valid for both $k = (1,5)$. Duncan Moore found only one primitive solution, namely, $$85333^k +...

Let $k$ be a commutative ring and let $A$ be a commutative $k$-algebra. The cotangent complex $\mathbf{L}_{A \mid k}$ can be computed using a simplicial resolution of $A$ as follows: choose a simplicial commutative $k$-algebra $P$ and an augmentation $\epsilon : P_0 \to A$ (i.e. a $k$-algebra...

Vakil, Definition 20.1.1: Let $X$ be a variety (reduced separated finite type scheme, actually I'm not sure if we need all this. I think we can get away with dropping "reduced" and "separated" and just assume $X$ is a finite type scheme) over a field $k$ (not necessarily algebraically closed)....

The trace form $(a,b)\mapsto \mathrm{tr}(ab)$ is easily motivated as a choice of bilinear form on a number field $K$ by noting that it agrees with the (standard real) dot product on $\mathbb{R}^{r_1}×\mathbb{C}^{r_2}$ as restricted to the Minkowski embedding of $K$ (the one that sends a number...