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The Ramanujan-Nagell Theorem: Understanding the Proof

The pdf version is A crash introduction to BSD conjecture . We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in $latex {\mathop{\mathbb P}^2}&fg=000000$. Definition 1 (Weierstrass form) $latex {E \hookrightarrow \mathop{\mathbb P}^2 }&fg=000000$, In general the form is given by, $latex \displaystyle E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)&fg=000000$ If $latex {char F \neq 2,3}&fg=000000$, then, we have a much more simpe…

The pdf version is A crash introduction to BSD conjecture . We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in $ {\mathop{\mathbb P}^2} $. Definition 1 (Weierstrass form) $ {E \hookrightarrow \mathop{\mathbb P}^2 } $, In general the form is given by, $  E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)$ If $ {char F \neq 2,3} $, then, we have a much more simper form, $ y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2) $ Remar…

A pdf version is Dirichlet hyperbola method. 1. Introduction Theorem 1 $\sum_{1\leq n\leq x}d(n)=\sum_{1\leq n\leq x}[\frac{x}{n}]=xlogx+(2\gamma-1) x+O(\sqrt{x}) \ \ \ \ \ (1)$ Remark 1 I thought this problem initial 5 years ago, cost me several days to find a answer, I definitely get something without the argument of Dirchlet hyperbola method and which is weaker but morally the same camparable with the result get by Dirichlet hyperbola method. Remark 2 How to get the formula: $\sum_{1\leq n\le…

I explain some general ideal in the theory of diophantine approximation, some of them is original by myself, begin with a toy model, then consider the application on folklore Swirsing-Schmidt conjecture. 目录 Contents 1. Dirichlet theorem, the toy model The very basic theorem in the theory of Diophantine approximation is the well known Dirichlet approximation theorem, the statement is following. Theorem 1 (Dirichlet theorem) for all ${\alpha}$ is a irrational number, we have infinity rational numb…

There is some gap, in fact I can improve half of the argument of Discrete harmonic function , the pdf version is Discrete harmonic function in Z^n, but I still have some gap to deal with the residue half… 1. The statement of result First of all, we give the definition of discrete harmonic function. Definition 1 (Discrete harmonic function) We say a function $ {f: {\mathbb Z}^n \rightarrow {\mathbb R}}$ is a discrete harmonic function on $ {{\mathbb Z}^n}$ if and only if for any $ {(x_1,&#8…

  This is a note concentrate on the log average Sarnak conjecture, after the work of Matomaki and Raziwill on the estimate of multiplication function of short interval. Given a overview of the presented tools and method dealing with this conjectue.   1. Introduction Sarnak conjecture [1] assert that for any obersevable $latex {\{f(T^n(x_0))\}_{n=1}^{\infty}}&fg=000000$ come from a determination systems $latex {(T,X),T:X\rightarrow X}&fg=000000$, where $latex {h(T)=0}&fg=000…

The story of the infinite dimensional space of $\Delta$ is following, we eliminate ourself with compact smooth non-boundary manifold $M$ with metric $g$, then we have Betrami-Laplace operator $\Delta_g$. We could instead $\Delta_g$ by hodge laplace $dd^*+d^*d$, but let we consider $\Delta_g$ the eigenvalue problem: $$\Delta_g u=\lambda u$$ A classical way to investigate the eigenvalue problem is according to consider variational principle and max-min principle. We equip the path integral on the …

The pdf version is Uncertainty principle. The nice note of terrence tao seems given a nice answer for the problem below. 1. Introduction Is there a Brunn-Minkowski inequality approach to the phenomenon charged by uncertainty principle? More precisely, is it possible to say some thing about the Gaussian distribution $ G(x)=e^{-|x|^2} \ \ \ \ \ (1) $   to be the best choice that $ {\|\hat G-G\|_2} $ arrive minimum? Remark 1 Or some other suitable distance space on reasonable function (ma…

In this short note, I posed a conjecture on Brunn-Minkwoski inequality and explain why we could be interested in this inequality, what is it meaning for further developing of some fully nonlinear elliptic equation come from geometry. The main part of the note devoted to discuss several different proof of classical Brunn-Minkowski inequality. Brunn-Minkowski inequality 1. Introduction I believe, every type of Brunn-Minkowski inequality, type of Brunn-Minkowski inequality is in some special sense …