The simplest definition of number theory is a branch of mathematics that deals with the study of real numbers or integers. Unlike other branches of math, concepts and problems in the number theorem can be understood by most people who do not have advanced mathematical capabilities. This is not to say that there aren’t theorems, and concepts do not require sophisticated mathematical backgrounds.
Looking at the number theory as a branch of pure mathematics, the study does not appear to have real-life applications in the real world. This perception was thrown out of the window when the concepts started being used in digital computers and programming where numerical problems were solved using number theorem and the solutions used in real-life situations using digital technology.
Computer inventions and advancement have a working symbiotic relationship, whereas number theory phenomenon was the basis for computer software programming, computer technology has allowed mathematical theorists to make significant steps by enabling them to be able to work with large numbers providing them with the means to test problems and concepts which they could otherwise not be able to without digital technology. The following subheadings are examples of the categories that will provide a basic introduction for beginners.
This is a branch where primary methods like high school algebra and geometry are used to solve equations with integers. The classification of Pythagorean triples is a good example of a problem which is solved using elementary theory. The Pythagorean triple is the theorem that relates positive integers a, b, and c in such a way that if a right triangle exists with legs a and b, and a hypotenuse c, then the sides will be governed by the relationship that has to satisfy that;
a2 + b2 = c2
Algebraic number theory is the branch that deals with algebraic numbers. This branch was developed as a set of logical tools that would be used to solve problems in elementary. The most common being Diophantine equations. These are equations that include integers and rational numbers that can be lifted from natural numbers and expressed as algebraic expressions.
Algebraic number theory also involves the study of the roots of polynomials with rational and integral coefficients. It envisions numbers lying in algebraic structures that have properties that are similar to those of integers. The most critical conjecture in an algebraic number system is Fermat’s conjecture that was eventually proved in the 1990s.
Analytic Number Theory
This is the branch that uses real and complex analysis to investigate the common properties of integers and prime numbers deeply. There are no defined preconditions of the study of analytic number theory, but it majorly focuses on the analytic activity of determining and estimating objects and handling error terms. Some underlying assumptions within the analytic include the Riemann zeta function, arithmetic progression and functions like the Dirichlet functions and basic distributional principles and functions.
Geometric Number Theory
Geometric closely related to algebraic number theory. It involves the study of polynomials over fields of number-theoretic interest. Theories like the Diophantine geometry of numbers is part of these geometric number theories.
Other theories include Galois representations, Langland’s programs, the Artine-Verdier theory of duality and many other complex phenomena that represent integers and real numbers using algebraic and geometric expressions to simplify their logical solutions.
This is the use of probability techniques to study the behavior and relationship of integers and L-functions. Number theory techniques have also proven to be effective in resolving problems in discrete probability theory.
These two schools of thought connect in the study of when the probability is used to create models for integers when probability techniques are used to prove results, when number theory problems lead to questions of probability and when number theory methods are used to determine probability results definitively. Probability methods and phenomena are dependent on each other.
In number theory, the relationship between the additive and multiplicative structures for integers has fascinated mathematicians, and they have made number theory into a wide field of mathematical research. The phenomena involved are quite complicated, but if one has the interest of pursuing deeper into the number theory he can quickly get help by going through the studies doe and tabulated by mathematicians like Gauss.
These studies are found in scholarly articles all over the internet. You can also benefit from the works done by Euclid and Pythagoras. They are known to have established relationships that are fundamental to the understanding.
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