1615905600
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). In this number theroy course following topics hav been explained in a very comprehensive way.
⭐️ Table of Content ⭐️
⌨️ (00:00) Introduction to number theory
⌨️ (02:07) The principle of mathematical induction
⌨️ (06:04) Basic representation theorem
⌨️ (10:14) The division algorithm
⌨️ (13:49) The divisibility
⌨️ (17:31) The euclidean algorithm
⌨️ (22:48) Linear Diophantine Equations
⌨️ (26:03) The fundamental theorem of arithemetic
⌨️ (30:38) Permutations and combinations
⌨️ (36:10) Fermat’s Little theorem
⌨️ (38:51) Wilson’s Theorem
⌨️ (42:42) Computer Programming
⌨️ (49:18) Basic properties of congruences
⌨️ (52:51) Residue Systems
⌨️ (58:12) Linear Congruences
⌨️ (1:01:54) Fermat’s little theorem and wilson’s theorem
⌨️ (1:06:31) The Chinese remainder theorem
⌨️ (1:10:50) The Eular Phi Function Part 1
⌨️ (1:14:34) The Eular Phi Function Part 2
⌨️ (1:19:12) Multiplicative function
⌨️ (1:23:51) The mobious inversion formula
⌨️ (1:28:08) Order of Elements
⌨️ (1:33:49) Primitive roots modolo
⌨️ (1:37:03) The prime counting function
⌨️ (1:41:39) The Eular’s criterion
⌨️ (1:45:44) The Legendre symbol
⌨️ (1:48:42) Quadratic Reciprocity part 1
⌨️ (1:53:17) Quadratic Reciprocity part 2
⌨️ (1:59:33) Application of quadratic reciprocity
⌨️ (2:02:13) Consicutive Residues
⌨️ (2:06:54) Consicutive triples of Residues part 1
⌨️ (2:09:03) Consicutive triples of Residues part 2
⌨️ (2:13:40) Sums of two squares
⌨️ (2:16:02) Sums of four squares
⌨️ (2:22:16) Gauss circle problem
⌨️ (2:25:38) Dirichlet’s devisor problem
⌨️ (2:29:55) Infinity Conclusion
#developer
1630996646
Class static blocks provide a mechanism to perform additional static initialization during class definition evaluation.
This is not intended as a replacement for public fields, as they provide useful information for static analysis tools and are a valid target for decorators. Rather, this is intended to augment existing use cases and enable new use cases not currently handled by that proposal.
Stage: 4
Champion: Ron Buckton (@rbuckton)
For detailed status of this proposal see TODO, below.
Motivations
The current proposals for static fields and static private fields provide a mechanism to perform per-field initialization of the static-side of a class during ClassDefinitionEvaluation, however there are some cases that cannot be covered easily. For example, if you need to evaluate statements during initialization (such as try..catch), or set two fields from a single value, you have to perform that logic outside of the class definition.
// without static blocks:
class C {
static x = ...;
static y;
static z;
}
try {
const obj = doSomethingWith(C.x);
C.y = obj.y
C.z = obj.z;
}
catch {
C.y = ...;
C.z = ...;
}
// with static blocks:
class C {
static x = ...;
static y;
static z;
static {
try {
const obj = doSomethingWith(this.x);
this.y = obj.y;
this.z = obj.z;
}
catch {
this.y = ...;
this.z = ...;
}
}
}
In addition, there are cases where information sharing needs to occur between a class with an instance private field and another class or function declared in the same scope.
Static blocks provide an opportunity to evaluate statements in the context of the current class declaration, with privileged access to private state (be they instance-private or static-private):
let getX;
export class C {
#x
constructor(x) {
this.#x = { data: x };
}
static {
// getX has privileged access to #x
getX = (obj) => obj.#x;
}
}
export function readXData(obj) {
return getX(obj).data;
}
The Private Declarations proposal also intends to address the issue of privileged access between two classes, by lifting the private name out of the class declaration and into the enclosing scope. While there is some overlap in that respect, private declarations do not solve the issue of multi-step static initialization without potentially exposing a private name to the outer scope purely for initialization purposes:
// with private declarations
private #z; // exposed purely for post-declaration initialization
class C {
static y;
static outer #z;
}
const obj = ...;
C.y = obj.y;
C.#z = obj.z;
// with static block
class C {
static y;
static #z; // not exposed outside of class
static {
const obj = ...;
this.y = obj.y;
this.#z = obj.z;
}
}
In addition, Private Declarations expose a private name that potentially allows both read and write access to shared private state when read-only access might be desireable. To work around this with private declarations requires additional complexity (though there is a similar cost for static{} as well):
// with private declarations
private #zRead;
class C {
#z = ...; // only writable inside of the class
get #zRead() { return this.#z; } // wrapper needed to ensure read-only access
}
// with static
let zRead;
class C {
#z = ...; // only writable inside of the class
static { zRead = obj => obj.#z; } // callback needed to ensure read-only access
}
In the long run, however, there is nothing that prevents these two proposals from working side-by-side:
private #shared;
class C {
static outer #shared;
static #local;
static {
const obj = ...;
this.#shared = obj.shared;
this.#local = obj.local;
}
}
class D {
method() {
C.#shared; // ok
C.#local; // no access
}
}
Prior Art
Syntax
class C {
static {
// statements
}
}
Semantics
Examples
// "friend" access (same module)
let A, B;
{
let friendA;
A = class A {
#x;
static {
friendA = {
getX(obj) { return obj.#x },
setX(obj, value) { obj.#x = value }
};
}
};
B = class B {
constructor(a) {
const x = friendA.getX(a); // ok
friendA.setX(a, x); // ok
}
};
}
References
TODO
The following is a high-level list of tasks to progress through each stage of the TC39 proposal process:
For up-to-date information on Stage 4 criteria, check: #48
Download Details:
Author: tc39
The Demo/Documentation: View The Demo/Documentation
Download Link: Download The Source Code
Official Website: https://github.com/tc39/proposal-class-static-block
License: BSD-3
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