Let $S$ be a Dedekind scheme (using the less conventional definition: locally Noetherian, irreducible, all stalks normal and $\dim S \le 1$), $X$ an irreducible scheme, and $f: X \to S$ a dominant morphism of finite type. Consider a point $s \in S$, and let $T = \operatorname{Spec}...

I try to understand what the Grassmannian $\operatorname{Gr}(2, F^3)$ (over some field $F$) looks like geometrically (in as far this term makes sense when we do not specify the field), and in particular how two subsets (specified below) divide the whole thing among them. To my shame I do not...

Semiprime Covering of Residue Classes Modulo a Primorial Let $p$ and $q$ be consecutive primes, with $q$ the smallest prime greater than $p$, and let $p\#=\prod_{\ell\le p,\ \ell\text{ prime}}\ell$ denote the primorial of $p$. Let $\mathcal S$ denote the set of semiprimes, $\mathcal...

This is in regards to building the DAG and showing the structural organization of the system rather than the superficial view of the Syracuse map

Let $S$ be a K3 surface over some field $k$. That is, $S$ is a nice variety over $k$ such that the canonical bundle $\omega_S$ is trivial and $H^1(S, \mathcal{O}_S) = 0$. As an exercise for myself, I wanted to see if I can compute the Hodge numbers $h^{p,q} := \dim_k H^q(S,\Omega^p_S)$. I have...

Let $p$ be an odd prime and $a\geq 1$. Consider the reduced residue system modulo $p^a$, namely the set of integers $x$ such that $$ 1\leq x\leq p^a,\qquad \gcd(x,p)=1 $$ I would like to determine the following power sum modulo $p^a$: $$ S_k=\sum_{\substack{1\leq x\leq p^a\\ \gcd(x,p)=1}}x^k...

Are there injections $\phi:SL_2(\mathbb{Z})\to M^{n\times n}(\mathbb{Z})$ that decrease the size of the coefficients? For the application I have in mind, coefficient growth is exponential in the word length w.r.t. a generating set, e.g. the maximum entry of words of length $n$ in the generators...

(I'm sorry for my English.) Hello. I have been reading the book Advanced Topics in the Arithmetic of Elliptic Curves written by Joseph H. Silverman. In the course of the proof of Tate's algorithm (page 375, at the end of the proof of Step 8), there is an equality on the order as follows:...

It is a well-known fact that there exists a non‑principal ultrafilter $\mathcal{U}$ on the set of prime numbers such that the ultraproduct of the algebraic closures of the finite fields is isomorphic to the complex numbers: $$ \mathbb{C}\;\cong\; \prod_{\mathcal{U}}\ \overline{\mathbb{F}}_p . $$...

I have a question regarding this paper by Tenenbaum and Bréteché. They define $$ S_\alpha(\mathcal{M}) = \sum_{m,n\in\mathcal{M}} \frac{(m,n)^\alpha}{[m,n]^\alpha} = \sum_{m,n\in\mathcal{M}} \biggl( \frac{(m,n)^2}{mn} \biggr)^\alpha \quad\text{and}\quad \Gamma_{\alpha}(N) =...

$\def\Z{{\Bbb Z}}\let\f\frac\let\d\delta$Concerning binary forms, there is something I don’t understand: Let $F$ be a form $F=aX^2+2bXY+cY^2$, $a,b,c\in\Z$ or $\Z_2$. Let assume $F$ to be primitive, that is $a$ or $c$ odd, and so invertible (or unit) in $\Z_2$. I write $$F=a\left(X+\f...

List of Mersenne prime exponents: [2,3,5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583,...

Consider the Diophantine equation: $$x_1^k+x_2^k+\dots+x_n^k=y^k.\tag 1$$ For any exponent $k$, let $N(k)$ be the smallest possible number of terms on the left, $n$, such that there exists a non-trivial solution in positive integers. Euler’s Sum of Powers Conjecture is equivalent to the...

It is well known that a parameterization for the equation $$W^2+X^2=Y^2+Z^2$$ is given by: $$(W,X,Y,Z)=(a+b,ab-1,ab+1,a-b).$$ I have found a similar type of parameterization for the fourth-power equation: $$W^4+X^4=Y^4+Z^4$$ where $W = d+e$, $X = cde-1$, $Y = d-e$, and $Z = cde+1$, provided that...

Let $\Phi=[F_1,\ldots,F_N]$ be a map, with $F_i$ homogenous polynomials of variables $X_1,\ldots,X_N$ of the same degree with integer coefficients. The map $\Phi$ mapping $Z^N$ into itself may share the following property: Let $P=(x_1,\ldots,x_N)\in Z^N$ be any point such that $\gcd...

I have zero, one, addition, subtraction, multiplication, division (by a non-zero expressible number), n-th roots (of positive expressible numbers, where n is expressible), and all the trigonometric functions (in their default domain restricted to the expressible numbers). The number expressible...

I'm a high school freshman with a strong passion for mathematics. I'm currently studying number theory through a math circle, and I'm particularly interested in pursuing mathematical research in areas such as Egyptian fractions, prime numbers, Catalan numbers, and related topics. I've been...

Let $a,d$ be positive integers. Is there a reasonably short elementary proof that one can find distinct nonnegative integers $n_1,\dots,n_k$ such that $$ \frac1{a+n_1d}+\frac1{a+n_2d}+\cdots+\frac1{a+n_kd}=1? $$ Equivalently, can one choose finitely many distinct terms of every infinite...

I am self studying Algebraic geometry from Gortz and Wedhorn's Algebraic Geometry :1 Schemes. I have a question on Page $16$ of the textbook just after the definition of morphism of affine algebraic sets. Remark $1.29$: the definition (of morphisms between affine algebraic sets) shows that a...

I am unable to understand the proof of proposition $1.40$ given on Page $21$ of the textbook by Gortz and Wedhorn. Definition $1.30$ Let $X\subset \mathbb{A}^n{k}$ be the affine algebraic set The $k-$algebra $\Gamma(X)= k[T_1,...,T_n]\cong Hom (X, \mathbb{A}^1(k))$ is called the affine...