Number theory is called the branch of pure mathematics which involves the study of numbers and integers. It involves understanding the relationship between numbers and especially positive integers. It may be easy to start learning number theory initially but once you dive deeper inside it, one may require sophisticated tools to gain in-depth knowledge in it.

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Number Theory is the challenging and the oldest branch of Mathematics. It offers a variety of career opportunities for those who want to make a career in mathematics. Multiple industries offer career opportunities in number theory with multiple ranges of tasks. Let us explore a few opportunities in different sectors.

Information Technology: "IT" refers to the use of computer hardware, software, and networking technology to create, store, and use data. A career as a software developer is available in information technology. The job of a software developer is to use creative and logical thinking to create software programs that meet the needs of software users. Another career option is to become a computer research scientist, whose job is to conduct extensive research and devise novel solutions to information technology-related problems.

Business: Those who enjoy working with numbers and can easily understand their relationships can start businesses that involve forecasting, resource allocation, or work for financial institutions. Accountants play a critical role in the execution of financial operations, the computation of taxes, and the completion of financial records for businesses. One can also become an economist, whose job is to forecast data for the future to assist businesses in making sound decisions.

Science:

Science is the study of numbers to reach conclusions. One can pursue a career as a microbiologist or physicist in this field. Microbiologists study algae, bacteria, viruses, and other forms of life. After conducting experiments, they collect data to determine how these organisms live and how they can benefit society. Physicists study the workings of nature, such as gravity, atomic structures, energy, and so on. They gain a better understanding of the world by studying fundamental properties of space, matter, and so on.

Engineering:

Engineering involves playing with lots of numbers. Engineering entails a lot of number-crunching. The job of an engineer is to design and build things. A career as a biomedical engineer or a civil engineer is possible. A biomedical engineer studies biological sciences, engineering principles, and software to create a variety of products ranging from artificial organs to diagnostic equipment. A civil engineer, on the other hand, is responsible for the construction of various infrastructures such as roads, bridges, airports, water systems, and so on. They hold positions in both the public and private sectors. Civil engineers analyse people's long-term needs and design specific plans, which include working on cost estimates, which are heavily reliant on a strong understanding of number analysis.

Data Science:

Data science is the field that involves the study of real-world trends and behaviour. This field is used to enhance business operations. Career opportunities in the data science field are statisticians, market research analysts, information research scientists, etc.A statistician’s job is to generate and analyse data to form conclusions and make decisions. Decision-making involves forming wise decisions regarding business techniques improvisation. Market research analyst job is to study market conditions, ups and downs to assist businesses to make decisions regarding sales and services. They find effective strategies for business success. These strategies result from thorough market research.

Number Theory is one of the most important fields of Mathematics. Number Theory mainly involves the study of whole numbers and positive integers. To gain in-depth knowledge of any subject, understanding basic concepts is very important. We become experts when the core is strong. The more common concepts of Number Theory are Primes and prime factorization, square numbers, prime numbers, rational numbers, divisibility of numbers, greatest common divisors, etc.

Number Theory specially deals with solving numerous equations. Most of the part of Number Theory is derived from Geometry. To answer Number Theory questions, one must be familiar with the concepts of differentiation, integration, infinite series, etc. The study of Number Theory that deals with the analysis of numbers is “Analytic Number Theory”. The other branch of Number Theory is Algebraic Number Theory which uses concepts like groups, rings, number fields, Galois theory which are the concepts of modern algebra.

The most common theorems and concepts in Number Theory are the Fundamental Theorem Of Arithmetic And The Division Algorithm, Euclid's theorem, GCD, LCM, Bezout's Identity, Linear Congruence Equations, Chinese Remainder Theorem, Quadratic Congruences, Integer Sequences, etc. The Fundamental Theorem Of Arithmetic And The Division Algorithm forms the heart of all the concepts in number theory.

Understanding the base of these concepts at an early stage is very important for the students who want to pursue a career in the subject. The concepts that are mentioned above, have their applications in fields like Information technology, business, network security systems, and so on.

Number Theory is a branch of pure Mathematics which itself is a vast concept. Number Theory is also called “Higher Arithmetic”. It is a subject that fascinates mathematicians. Number Theory is capable of providing answers to real-life problems. Advancement in Computer technology has brought advancement in concepts of Number theory like determining prime, testing conjectures, factoring large numbers, solving numerical problems, etc.

Number Theory is further divided into multiple sub-topics such as algebraic Number Theory, geometric number theory, analytic number theory, probabilistic number theory, and elementary number theory.

Algebraic Number Theory:It is the branch of number theory that involves the study of integers, rational numbers, generalisations by using techniques of abstract algebra. Applications of Number Theory can be seen in the fields of statistics and design theory.

Geometric Number Theory:It forms close relationships with other fields of Mathematics such as functional analysis, rational numbers, Diophantine approximation, etc. It involves mainly the study of algebraic numbers with the help of geometry. Geometric number theory is mainly used in telecommunications, coding theory, etc.

Analytic Number Theory:It is the branch of Number Theory that uses mathematical analysis methods to solve integer-related problems. Understanding analytic number theory, one should know complex analysis. Analytic number theory is used in the field of cryptography and its other related areas.

Probabilistic Number Theory:As the name suggests, it is the sub-branch of Number Theory that uses probability to solve problems about integers and their functions. Probabilistic number theory uses probabilistic tools to get the results. It is the branch of Number Theory that is evolving fast.

Elementary Number Theory:Elementary Number Theory is used to solve equations with integers and rational numbers with the use of elementary methods like algebra, geometric, arithmetic, etc. Applications of elementary number theory can be seen in frequency-hopping radar, cryptography, wireless communication, etc.

If you are fascinated by numbers and like to play with them, then Number Theory is the subject you must study. Number Theory helps us understand the relationships between different numbers. It provides valuable training in logical thinking. Let us understand the different fields where Number Theory applications can be seen.

Cryptography:

Cryptography is all about concealing sensitive information or converting it into an unreadable format. It makes use of encryption technology. Primes, divisors, congruences, and Euler's function, among other Number Theory tools, are widely used in cryptography. These tools are used to ensure security. The goal of cryptography is to send and receive encrypted messages. Cryptography is used to protect credit cards, debit cards, and bank records from people who should not have access to them. Cryptography secures data transmission over public communication networks such as satellites and telephone lines.

Device Authentication(RSA Cryptosystem):

RSA is a widely used public-key cryptosystem. RSA was invented by Rivest, Shamir, and Adleman. The main concept of RSA is based on the complexity of handling large primes, and the security is based on difficulty in the computation of factoring large numbers. In short, the algorithm's security lies in the challenge of prime factorization. The keys generated in the RSA system help authenticate devices as both the devices are aware of the keys and their encryption.

Coding:

Number Theory also plays a vital role in coding. Let’s explore how?

Linear algebra is used in game development, graphics design, animation, and stuff. Application of integral and differential calculus can be seen in signal processing applications. If you are into competitive programming, then probability, permutation, and combination play a vital role. Statistics can be used in a data management type of programming.

E-commerce website:

As we have seen over the years, the growing e-commerce and internet industry has resulted in huge amounts of data gathering. This data can include sensitive information like financial transactions, personal financial information like credit cards, etc. Also, the increase in the number of transactions happening in an e-commerce website has made it necessary that the data that is being used should be secured (encrypted) both in transit and at rest. The concepts of Number Theory have played a vital role in implementing concepts like cryptography which helps in data security in transit and at rest.

Number Theory is a branch of mathematics that studies the relationships between numbers and integers. Number theory is a broad concept with sub-categories such as algebraic number theory and analytic number theory. A strong background in algebra, Commutative algebra, algebraic geometry, algebraic topology, and so on is required to become a master of Number Theory. So, while it is not an easy subject for the majority of students, they cannot avoid Number Theory assignments. Failure to complete assignments on time may result in students losing grades in the subject. So the question is, how can students complete Number Theory assignments on time and earn good grades in the subject despite all of the difficulties? We have exactly what you are looking for.

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The best way to learn Number Theory is to have someone knowledgeable teach you - either in a classroom or individually. The next thing would be to gather a group of people who share your interest and teach yourselves through regular discussion meetings.

If neither of the options above is available to you, and you are on your own, then the best way is to start studying Number Theory by an introductory textbook that moves at a pace that would be right for you. There are numerous introductory textbooks on Number Theory that cover topics in an elementary manner and require no prior knowledge beyond some high school topics. Elementary Number is a book that has grown in popularity in India over the years.

There are multiple editions of this book. The most recent version adequately introduces the fundamental topics in Number Theory. The author straightforwardly explains the topics in a straightforward manner, covering all of the essentials. There is a vast array of exercises available, beginning with the routine and the simple. The only thing is, the book does not have enough challenging exercises for the reader. This book contains a couple of topics that are not found in any other introductory textbook, so that is a plus.

It is not enough to passively read the book, as it is with any other branch of Mathematics. You must always be prepared to take notes on special points such as definitions, examples, statements of all results, etc. After reading the proof of a particularly significant result, you must attempt to recreate the proof in your own words. Number Theory, in particular, necessitates that you supplement your understanding of the concepts by working on as many exercises as time and energy allow. In Number Theory, it is the variety of problems you work on, not the number of exercises you try. Having said that, there are many similar-looking problems in elementary mathematics.

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