Saturday, May 18, 2019

Number theory

The take system concerns about sum ups i.e. whole numbers or rational numbers (fractions). subdue system 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 to a fault 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 ar various number theories set forth as followsElementary Number theory Analytic Number theory Algebraic Number theory Geometric number theory Combinational number theory Computational number theory FUNCTIONSNumber theory is connected with higher arithmetic hence it is the study of properties of whole numbers. primals and prime factorization atomic number 18 important in number theory. The departs 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 ill-judged numbers. The study of irrational numbers may be done with Surd, Extraction of Square roots of natural numbers, Logarithms and Mensuration.At have Number Theory functions have 848 formulas, which are related with Prime Factorization Related functions and Other Functions.Prime Factorization Related FunctionsFactor integer n 70 FormulasDivision n 66 FormulasPrime n 83 FormulasPrimePi x 83 FormulasDivisor Sigma k,n 128 FormulasEuler Phi n 109 FormulasMoebius Mu n 79 FormulasJacobi Symbol n,m 101 FormulasCarmichasel Lambda n 63 FormulasDigit Count n, b 66 FormulasComputational 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 general 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 figuring or iteration. It is also study of many specific families of numbers like binomial coefficients, the Fibonacci numbers, Bernoulli numbers, factorials, perfect squares, breakdown numbers etc. which can be obtained by simple recurrence relations. The method is very escaped 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 multinomial in the length of p and q.Discrete logarithm Let G be a conference in which computations are reasonably efficient. Then given g and n, computing gn is not too exp ensive. However, for several(prenominal) groups G, computing n given g and gn, called the discrete logarithm, is difficult. The commonly used groups areDiscrete logarithms modulo p Elliptic curve discrete logarithms REFERENCEhttp// 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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