Number theory

Published: 2021-08-18 05:10:07
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Category: Mathematics, Algebra, Theories, Geometry

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The number theory concerns about numbers i.e. whole numbers or rational numbers (fractions). Number theory is one of the oldest branches of pure mathematics and one of the largest. It is a branch of pure mathematics concerning with the properties and integers. Arithmetic is also used to refer number theory. It is also called higher arithmetic. The earliest geometric use of Diophantine equations can be tracked back to the Sulba Sutras, which were written, between 8th and 6th centuries BC. There are various number theories described as follows:

Elementary Number theory

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Analytic Number theory
Algebraic Number theory
Geometric number theory
Combinational number theory
Computational number theory

Number theory is connected with higher arithmetic hence it is the study of properties of whole numbers. Primes and prime factorization are important in number theory. The functions in number theory are divisor function, Riemann Zeta function and totient function. The functions are linked with Natural numbers, whole numbers, integers and rational numbers. The functions are also linked with irrational numbers. The study of irrational numbers may be done with Surd, Extraction of Square roots of natural numbers, Logarithms and Mensuration.
At present Number Theory functions have 848 formulas, which are related with Prime Factorization Related functions and Other Functions.
Prime Factorization Related Functios
Factor Integer [n]                                          70 Formulas
Division [n]                                                    66 Formulas
Prime [n]                                                        83 Formulas
PrimePi [x]                                                     83 Formulas
Divisor Sigma [k,n]                                       128 Formulas
Euler Phi [n]                                                  109 Formulas
Moebius Mu [n]                                             79 Formulas
Jacobi Symbol [n,m]                                      101 Formulas
Carmichasel Lambda [n]                               63 Formulas
Digit Count [n, b]                                          66 Formulas
Computational number theory
It is a study of effectiveness of algorithms for computation of number-theoretic quantities. It is also considers integer quantities (for example class number) whose usual definition is non constructive, and real quantities (eg. The values of zeta functions) which must be computed with very high precision. Hence in this function overlaps both computer algebra and numerical analysis.
Combinational Number Theory
It involves the number-theoretic study of objects, which arise naturally from counting or iteration. It is also study of many specific families of numbers like binomial coefficients, the Fibonacci numbers, Bernoulli numbers, factorials, perfect squares, partition numbers etc. which can be obtained by simple recurrence relations. The method is very easy to state conjectures in this area, which can often be understood without any particular mathematical training.
Integer factorization
Given two large prime numbers, p and q, their product pq can easily be computed. However, given pq, the best known algorithms to recover p and q require time greater than any polynomial in the length of p and q.
Discrete logarithm
Let G be a group in which computations are reasonably efficient. Then given g and n, computing gn is not too expensive. However, for some groups G, computing n given g and gn, called the discrete logarithm, is difficult. The commonly used groups are
Discrete logarithms modulo p
Elliptic curve discrete logarithms
Weil, Andre: “Number theory, An approach through history”, Birkhauser Boston, Inc. Mass., 1984 ISBN-0-8176031410
Ore, Oystein, “Number theory and its history, Dover Publications, Inc., New York, 1988. 370 pp. ISBN 0-486-65620-9.

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