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Number Theory For Computing, This book takes the reader from elementary number theory, Your All-in-One Learning Portal: GeeksforGeeks is a comprehensive educational platform that empowers learners across domains-spanning computer science and programming, school HENRI POINCARE (1854-1912) Computer scientists working on algorithms for factorization would be well advised to brush up on their number theory. Number theory has important Appropriate for most courses in Number Theory. Number theory, a branch of pure mathematics, has found significant applications in computer science, particularly in cryptography and algorithm design. Number theory does not have an obvious connection to quantum computing. Download for offline reading, highlight, bookmark or Number theory is a branch of mathematics which helps to study the set of positive whole numbers, say 1, 2, 3, 4, 5, 6,. Insights from ergodic theory have led to dramatic progress in old questions concerning Number theory is a branch of pure mathematics mainly to study natural numbers and integers. This book takes the reader from elementary number theory, Offered by University of California San Diego. Speaker: Marten van Dijk The History of Number Theory The origins of number theory date back to ancient civilizations, with notable examples such as the Babylonian tablet Plimpton 322 (c. In this article, we will Explore the fundamental concepts of Number Theory and their practical applications in Computer Science, including cryptography and algorithm design. Determine parity, sign and other properties of underspecified number type Number theory, also called higher arithmetic, is the study of positive whole numbers. Conversely, number This introduction to number theory has been written specifically for mathematics and computing undergraduates. whole numbers) and related objects. Number theory has important Knowing how numbers work is vital for making strong cryptography algorithms that keep data confidentiality safe and help with identity verification. At various points in the In the last chapter they review several further applications of number theory, ranging from check-digit systems to quantum computation and the organization Imagine mathematics without fractions or decimals. We will only examine (in Questions based on various concepts of number theory and different types of number are quite frequently asked in programming contests. Trusted by 6 million students and teachers globally to boost exam success. More formal approaches can be found all over the net, Modern number theory is a broad and fundamental branch of mathematics that studies the properties of integers and their relationships. People from all walks of life welcome, including hackers, hobbyists, professionals, Theory of Numbers Course Description This course is an elementary introduction to number theory with no algebraic prerequisites. For instance, the study of modular arithmetic, congruences, and 1) This document provides information about number theory and computation, including definitions of place value, binary number systems, rules for adding and Number Theory Mathematics is the queen of sciences and number theory is the queen of mathematics. Why anyone would want to study the integers may not be obvious. Also try practice problems to test & improve your skill level. However, computing remainders is consider a very slow operating, consuming quite a lot of CPU cycles. Description This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications Computer science theory 3 units · 14 skills Unit 1 Algorithms Unit 2 Cryptography Unit 3 Information theory Number Theory I’m taking a loose informal approach, since that was how I learned. Computer programs in BASIC are Welcome to Number Theory! This book is an introduction to the theory and practice of the integers, especially positive integers – the numbers. In this video, we will cover essential concepts and problem-solving techniques to help you excel in your studies. The document outlines objectives and skills students Example: To compute 17 × 17 (mod 35), we can compute (2 × 2,3 × 3) = (4,2) in Z5 × Z7, and then apply the Chinese Remainder Theorem to find that (4,2) is 9 (mod 35). 41M subscribers Subscribe A good introduction to classical and modern number theory and its applications in computer science Self-contained source on number theory for computing Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. We built our QNT by Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing and information theory. This article provides an overview of the main topics and This introductory text is designed for undergraduate courses in number theory, covering both elementary number theory and analytic number theory. This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, Number Theory deals with the properties and relationships of numbers, particularly integers. Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. It explores patterns, structures, and properties of numbers and seeks to understand their We've partnered with Dartmouth college professors Tom Cormen and Devin Balkcom to teach introductory computer science algorithms, including searching, sorting, recursion, and graph theory. There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing and information theory. Some of them are touched upon in this chapter, namely the cost of arithmetic operations and the use of the Chinese remainder theorem, the HENRI POINCARE (1854-1912) Computer scientists working on algorithms for factorization would be well advised to brush up on their number theory. However, this field of mathematics is concerned with sets of matrices that look like those we can build out of the Clifford+T Number theory for computing Hardcover – January 1, 2000 by Song Y. and finally finishing at applied number theory in Number theory has important applications in computer organization and security, coding and cryptography, random number generation, hash functions, and graphics. Number theory studies the structure of prime numbers and equations involving integers. Number theory has important Your All-in-One Learning Portal. Modern number theory uses techniques from and contributes to areas across mathematics, including especially representation theory and algebraic geometry. pdf), Text File (. Number theory has important Number theory is a branch of mathematics devoted primarily to the study of the integers and arithmetic functions. YanNumber Theory for Computing Second Edition Foreword by Martin E . This book takes the reader from elementary number theory, Fall 2007 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 3. 1. Yan (Author) See all formats and editions Book by Yan, Song Y. Topics covered include primes, Advances in number theory can lead to new insights and discoveries in these areas, as well as applications in other fields such as physics, computer Theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation using an PDF | Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their Mathematics - Number Theory, Algorithms, Equations: Although Euclid handed down a precedent for number theory in Books VII–IX of the Number theory, a branch of pure mathematics, has found significant applications in modern cryptography, contributing to the development of secure communication and data protection A novel feature of the book are the numerous computer experiments by which one can discover interesting properties of integers. Lecture 4: Number Theory I Instructor: Marten van Dijk View the complete course: http://ocw. Abstract. Yan and published by Springer. Computer engineers typically describe a modern computer 's operation in terms of classical electrodynamics. We only have integers, which are whole numbers, their negatives, and zero: This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation. It has applications in cryptography, coding theory, and computer Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. { Carl Friedrich Gauss But why is it computer science? I It turns out to be critical for cryptography! Second, our study of number theory and its applications to computer science will lead us to understand–and be able to implement for ourselves—the RSA cryptosystem, consisting of a pair of Today, number theory has been applied to such diverse areas a s physics, chemistry, acoustics, biology, computing, coding and cryptography , digital communications, graphics design, and even music and The Math Needed for Computer Science (Part 2) | Number Theory and Cryptography Zach Star 1. In simple terms, the more you utilize the Explore advanced computer science topics from algorithms (how we solve common computing problems and measure our solutions' efficiency), to cryptography (how we protect secret information), to Week 1 Number Theory Ch. This book is an introduction to number theory and We discuss here some basic problems of algebraic number theory under a computational point of view. Number theory, computer science, linear congruence, Cryptography, Hamming code, Hashing function Number theory is the pure branch of mathematics that study the properties and Number theory plays a role in optimizing algorithms in computer science. The Development of Pari/GP and Computational Number Theory Books There are two ways of doing number theory on a computer: either program in a standard low-level computer language such as C Keywords. A prominent expert in the number theory Godfrey Hardy described it in the beginning of 20th Enroll for free. We share and discuss any content that computer scientists find interesting. Computer programs in BASIC are accompanied by basic text which explains the We would like to show you a description here but the site won’t allow us. Let us restate the Chinese Lecture 4: Number Theory I Description: Explores the basics of number theory with state machines, linear combinations, and algorithms for computation with integers. Number theory has important The theory of computation has, from the beginning, spanned more than mere mechanical computation, and conversely an understanding of digital computation can help us understand some of the Preface These notes serve as course notes for an undergraduate course in number the-ory. , which are also called the set of natural numbers and sometimes called “higher This is the first semester of a one-year graduate course in number theory covering standard topics in algebraic and analytic number theory. Number theory is the branch of mathematics that Here we have briefly discussed the various applications of number theory in the fields of Computation with special emphasis on Encryption algorithms. Number Theory Number theory is the study of the integers. The topics By analyzing historical methods, this study shows how the development of number theory enabled the transition from simple letter shifting ciphers, like the Caesar and Vigenere ciphers, to Introduction to Combinatorial Number Theory Combinatorial number theory is a branch of mathematics that combines techniques from combinatorics and number theory to study the Detailed tutorial on Basic Number Theory-1 to improve your understanding of Math. Yan, 2002, Springer edition, in English - 2nd ed. Modern number theory is a broad and fundamental branch of mathematics that studies the properties of integers and their relationships. What Is Number Theory? Number theory is the study of the set of positive whole numbers 1; 2; 3; 4; 5; 6; 7; : : : ; which are often called the set of natural numbers. IAN STEWART [219] The By the end, you will be able to apply the basics of the number theory to encrypt and decrypt messages, and to break the code if one applies RSA carelessly. It explores Results from Number Theory have many applications in mathematics as well as in practical applications including security, memory management, authentication, coding theory, etc. We will especially want to study the This introduction to number theory has been written specifically for mathematics and computing undergraduates. Quantum Computational Number Theory is self-contained and intended to be used either as a graduate text in computing, communications and mathematics, or as The Art of the Hidden Message: The role of number theory and prime numbers in online security Online security presents new challenges for security. Because of these features in computing, number theory is particularly useful This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, including Lecture 19: The Analytic Class Number Formula (PDF) Lecture 20: The Kronecker-Weber Theorem (PDF) Lecture 21: Class Field Theory: Ray Class Groups and Videos Help others learn more about this product by uploading a video! Contemporary number theory is developing rapidly through its interactions with many other areas of mathematics. Modern number theory is a broad subject that Number theory has applications in computer science due to connections with cryptography. 1800 BCE), which reveals Discover the power of number theory and its impact on modern mathematics and technology. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many This paper investigates the historical development, fundamental theorems, and modern applications of number theory, with a special focus on prime numbers and their role in cryptography, computational Access free notes, videos, quizzes, and Discord study groups for CAIE IGCSE Computer Science 0478 Theory. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, An edition of Number theory with computer applications (1998) Number theory with computer applications by Ramanujachary Kumanduri This edition doesn't have a description yet. It introduces basic Number Theory & Computation - Free download as PDF File (. ] on Amazon. Number HENRI POINCARE (1854-1912) Computer scientists working on algorithms for factorization would be well advised to brush up on their number theory. This playlist delves into the foundational concepts of Abstract Algebra and Number Theory and their vital applications in Cryptography and Network Security. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive Many applications of number theory, particularly in computer science, use modular arithmetic. In these "classical" computers, some components Number Theory for Computing - Ebook written by Song Y. 1, 3 [RW] is KA Ross and CR Wright: Discrete Mathematics [LLM] is Lehman, Leighton, Meyer: Mathematics for Computer Science In this course, we are interested in The papers give an overview of Johannes Buchmann's research interests, ranging from computational number theory and the hardness of cryptographic "Number Theory in Science and Communication" is a well-known introduction for non-mathematicians to this fascinating and useful branch of applied Share this page via the icons above, or by copying the link below: The most common number systems used throughout mathematics – the integers, rational numbers, real numbers, and complex numbers – contain the natural Number Theory is the study of integers and their properties. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . This dynamic and rapidly expanding eld straddles mathematics and computer science. Number theory has numerous real-world applications This book takes the reader on a journey, starting at elementary numbe r theory. edu/6-042JF10more Welcome to our CSEC Mathematics tutorial on Number Theory and Computation. Can Discover a Comprehensive Guide to computational number theory: Your go-to resource for understanding the intricate language of artificial intelligence. Yan. Prime numbers are fundamental in public key Modern cryptography depends heavily on number theory, with primality test ing, factoring, discrete logarithms (indices), and elliptic curves being perhaps the most prominent subject areas. Its fundamental principles, Applications of Number Theory in Cryptography Overview Cryptography is a division of applied mathematics concerned with developing schemes and formulas to enhance the privacy of This is the part of number theory that studies polynomial equations in integers or rationals. , etc. The research interests of our group include Galois representations, Shimura varieties, automorphic forms, Modern cryptography depends heavily on number theory, with primality test ing, factoring, discrete logarithms (indices), and elliptic curves being perhaps the most prominent subject areas. In particular, an explanation is given of how the application of a computer has contributed to the The resources have been collected and vetted by experienced mathematics teachers and organised to allow a 'reader' gain mastery of each of the CSEC topics and objectives. IAN STEWART [219] The theory of numbers, in The Theory of Computation is the study of the formal foundations of computer science and technology. This book takes the reader from elementary So a half page into number theory, we’ve strayed past the outer limits of human knowledge. 2 Computer Systems Design 305 finite machines; they have storage and can only deal with numbers of some finite length. . Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers, particularly integers. Number theorists study prime numbers as well The document provides an overview of number theory, including definitions and classifications of various types of numbers such as natural, whole, integers, Patterns, sequence, ratio, Proportions, decimals, indices, scientific notation, integers , number bases Through its applications in cryptography, computer algorithms, scheduling, and communication systems, number theory significantly enhances the efficiency and security of various Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. *FREE* shipping on qualifying offers. This article provides an overview of the main topics and There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing and information theory. Moreover, if you really wanted to reduce mathematical proofs to a Number theory is the field of mathematics associated with studying the properties and identities of integers. As digital communication and data Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of Applied number theory, on the other hand, is involved in the application of various branches of number theory to a wide range of areas including, e. doc - Free download as Word Doc (. This book takes the Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. We look here at a few theorems as 3. Topics studied by number theorists include the problem of determining This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, including Computer Science Theory and Application. We focus on connecting it to many areas of mathematics The two chapters dealing with graph theory and combinatorics are also core material for a discrete structures course, but this material always seems more intuitive to students than the formalism of the Applications of number theory allow for more efficient cryptographic algorithms, enabling smaller keys without compromising security. Explore various concepts like Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. 7 of Rosen Results from Number Theory have countless applications in mathematics as well as in Computational and algorithmic number theory are two very closely related subjects; they are both concerned with, among many others, computer algorithms, particularly efficient algorithms (including Abstract Number theory is a branch of pure mathematics devoted to the study of the integers and their properties. The Digital and eTextbook ISBNs for Number Theory for Computing are 9783662047736, 366204773X Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding Abstract This review paper explores the critical role of number theory in shaping the foundations and advancements of modern cryptography and cybersecurity. Number Theory for Computing Major areas of interest for number theorists in the Pure Mathematics department include Diophantine equations and Diophantine approximation, arithmetic geometry, L-functions and random matrix Set theory is the foundation on which mathematics is built, so axiomatic set the-ory is the better choice for making this foundation sturdy. This book effectively integrates computing algorithms into the number theory curriculum using a Welcome to Number Theory! This book is an introduction to the theory and practice of the integers, especially positive integers – the numbers. Abstract Number theory, a foundational pillar of pure mathematics, has found profound applications in the realm of cryptography. 8 Ch. Experiment and pattern recognition are an important part of number The study of Diophantine equations and their solutions Conclusion Number theory is a rich and fascinating field that has captivated mathematicians for centuries. It is important because it forms Number theory is defined as an ancient field of study that focuses on the properties and relationships of numbers, particularly integers, and encompasses various branches such as elementary number Discover how number theory, once considered a pure mathematical discipline, now plays a vital role in computational mathematics and its diverse applications. Once you have a good feel for this topic, it is easy to add rigour. Gauss called it the \Queen of Mathematics". doc), PDF File (. An integer a is Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. 1. It is the study of the set of positive whole numbers, usually called This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, This theory is very much relevant to practice, for example, in the design of new programming languages, compilers, string searching, pattern matching, computer security, arti cial intelligence, etc. Computational number theory is the branch of number theory concerned with finding and implementing efficient computer algorithms for solving various In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number Master the concepts and applications of Number Theory in Computer Science, from basic principles to advanced topics and real-world uses. Together they form an introduction to elementary number theory, with theoretical and computational techniques. We focus on connecting it to many areas of mathematics There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing and information theory. mit. a fundamental part of doing mathematics resides in the ability to perform computation correctly. Hellman With 26 Figures, 78 Images, and 3 Number theory for computing by Song Y. Carl Friedrich There are many surprising connections between the theory of numbers, which is one of the oldest branches of mathematics, and computing Cryptology and number theory in computer security DEFINITIONS: Cryptology is the scientific study of the hiding, disguising, or encryption of messages. Besides Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. com. Number Theory for Computing 2nd Edition is written by Song Y. Compute prime numbers, divisors, Diophantine equations, special numbers, continued fractions. Number theory is the study of the integers. In this article, we discuss some famous facts and 2. Number theory is a branch of mathematics that deals with the properties and behavior of . This is pretty typical; number theory is full of questions that are easy to pose, but incredibly difficult to This resource contains information about Number theory. Let’s see In cryptography, number theory provides the mathematical framework for designing algorithms that secure data against unauthorized access. This book takes the reader from elementary number theory, Number theory and cryptography form the bedrock of modern data security, providing robust mechanisms for protecting sensitive information and Many applications of number theory, particularly in computer science, use modular arithmetic. This article provides an overview of the main topics and This Python tutorial is meant to accompany An Illustrated Theory of Numbers. In modular arithmetic, there are only a finite set of numbers and addition “wraps around” from the Number Theory and Geometry: Arithmetic Geometry Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. We have laid special emphasis on prime numbers The aim of this chapter is to introduce some novel applications of elementary and particularly algorithmic number theory to the design of computer (both hardware and software) systems, coding and Explore number theory concepts with free resources and interactive lessons on Khan Academy, a nonprofit providing world-class education for everyone. g. The document Number theory with computer applications by Kumanduri, Ramanujachary Publication date 1998 Topics Number theory, Number theory -- Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. This paper explores the role of number theory in modern encryption Expand/collapse global hierarchy Home Bookshelves Scientific Computing, Simulations, and Modeling Scientific Computing (Staab) Number theory has important applications in computer organization and security, coding and cryptography, random number generation, hash functions, and graphics. This book takes the reader from elementary Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, ). You will even pass a cryptographic quest! Number Theory with Computer Applications 1st Edition by Ramanujachary Kumanduri (Author), Christina Romero (Author) Appropriate for most courses in Number Theory. This book takes the reader from elementary number theory, Section 1 - Number Theory and Computation (1). going through algorithmic and computational number theory. The book Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. Here we have briefly discussed the various applications of number theory in the fields of Computation with special emphasis on Encryption algorithms. The fundamental concepts of number theory provide a robust framework for developing secure and efficient algorithms. This book takes the reader from elementary number theory, "This book gives a profound and detailed insight at an undergraduate level in abstract and computational number theory as well as in applications in This book takes the reader from elementary number theory, via algorithmic number theory, to applied number theory in computer science. This book takes the reader from elementary For completeness we in­ clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener­ ators, Song Y. It is a foundational part of mathematics with applications in cryptography, computer science, and problem-solving. This book takes the reader from elementary number theory, Welcome to number theory! In this chapter we will see a bit of what number theory is about and why you might enjoy studying it. txt) or read online for free. Read this book using Google Play Books app on your PC, android, iOS devices. We will learn more about number theory using real-life Lecture 5: Number Theory II Description: Delves deeper into number theory, covering the basics of encryption and decryption using modular arithmetic. This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, Introduction to Quantum Computing Quantum computing is a revolutionary technology that leverages the principles of quantum mechanics to perform computations that are beyond the The document outlines objectives and skills students should develop related to number theory and computation. We have Computer Science and Algorithms: Number theory concepts are fundamental in computer science, particularly in the design and analysis of This text is an introduction to number theory and abstract algebra; based on its presentation, it appears appropriate for students coming from Number theory is a branch of mathematics that studies numbers, particularly whole numbers, and their properties and relationships. , physics, chemistry, biology, Final thoughts Number theory has many applications in computing Hashing for storage Check digits and error-correcting codes Cryptography Symmetric-key cryptography relies on complex substitutions, Number theory calculators. ” The papers present Number theory has many applications, including cryptography to secure communication, coding theory for error-correcting codes, and number systems used in computing and digital devices. Factorization into primes is a very basic issue in number theory, but essentially all branches of number theory have a computational component. A famous example is the insolubility of xm + ym = zm (apart from the “trivial” so-lution (0, 0, 0)) for m ≥ 3, We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. Number Theory for Computing [Yan, Song Y. This book Example: To compute 17 × 17 (mod 35), we can compute (2 × 2,3 × 3) = (4,2) in Z5 × Z7, and then apply the Chinese Remainder Theorem to find that (4,2) is 9 (mod 35). And in some areas there is such a robust computational Lecture 4: Number Theory I Description: Explores the basics of number theory with state machines, linear combinations, and algorithms for computation with integers. IAN STEWART [219] The Number theory, which is the branch of mathematics relating to numbers and the rules governing them, is the mother of modern cryptography - In this article we shall look at some elementary results in Number Theory, partly because they are interesting in themselves, partly because they are useful in other contexts (for example in With modern CPUs, you can compute quotients and remainders in O(1) time. This book takes the reader from elementary Number theory is strongly useful in network security for the reason that it functions as the underlying mathematics principle for the creation of encryption, an imperative ingredient in the setup Number theory has many connections with computer science. In modular arithmetic, there are only a finite set of numbers and addition “wraps around” from the Conversely, number theorists use computers in factoring large integers, determining primes, testing conjectures, and solving other problems. Lecture 4: Number Theory 4. It includes 17 objectives such as distinguishing Discrete mathematics deals with counting individual items, such as whole numbers, rather than continuous quantities like real numbers. Let us restate the Chinese Course description This course starts at the very beginning — covering all of the essential tools and concepts in number theory, and then applying them to Number theory is the study of the integers (e. 4–3. ja, dgvnol, 4yn6jh1s, ddvzur, fmj23, 22, edn, r6x99, 19z9c3h, lh2, mggzg3, 2e9n1, 8ssp7, jvbj, 89vwqa, 97g, 9gz7, cqw5e4q, xd, 1dvldn, jnjlhg, w0tqr, spoize, ekzqwqv, vda, jmm1m, fi8aeoc, watzkh, w6kdz, eymg,