Number theory. verify that 0,1,2,2^2,2^3,.......,2^9 form a complete set of residues modulo 11

1 ANSWERS

1. Ali Abdullah
   

   Hi, Elementary number theory In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function and Euler's f function, integer sequences, factorials, and Fibonacci numbers all also fall into this area. Many questions in number theory can be stated in elementary number theoretic terms, but they may require very deep consideration and new approaches outside the realm of elementary number theory to solve. Examples include: * The Goldbach conjecture concerning the expression of even numbers as sums of two primes. * Mihailescu's theorem (formerly Catalan's conjecture) regarding successive integer powers. * The twin prime conjecture about the infinitude of prime pairs. * The Collatz conjecture concerning a simple iteration. * Fermat's Last Theorem (stated in 1637, but not proven until 1994) concerning the impossibility of finding nonzero integers x, y, z such that xn + yn = zn for some integer n greater than 2. The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem). [edit] Analytic number theory Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. The prime number theorem (PNT) and the related Riemann hypothesis are examples. Waring's problem (representing a given integer as a sum of squares, cubes etc.), the twin prime conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well. Proofs of the transcendence of mathematical constants, such as p or e, are also classified as analytical number theory. While statements about transcendental numbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integer coefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation, where one investigates "how well" a given real number may be approximated by a rational one. [edit] Algebraic number theory In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so-called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed—Galois theory, group cohomology, class field theory, group representations and L-functions—is that it allows one to recover that order partly for this new class of numbers. Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory. [edit] Geometry of numbers The geometry of numbers incorporates some basic geometric concepts, such as lattices, into number-theoretic questions. It starts with Minkowski's theorem about lattice points in convex sets, and leads to basic proofs of the finiteness of the class number and Dirichlet's unit theorem, two fundamental theorems in algebraic number theory. [edit] Combinatorial number theory Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdos is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field. See also arithmetic combinatorics. [edit] Computational number theory Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography. [edit] Arithmetic algebraic geometry See arithmetic geometry. [edit] Arithmetic topology Arithmetic topology developed from a series of analogies between number fields and 3-manifolds; primes and knots pointed out by Barry Mazur and by Yuri Manin in the 1960s. [edit] Arithmetic dynamics Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function.

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