On the number of prime factors of a finite arithmetical progression
On the number of prime factors of a finite arithmetical progression

An Introduction to Unitary Representations of Lie Groups
An Introduction to Unitary Representations of Lie Groups

On Number theory algorithms from Srividya and George
On Number theory algorithms from Srividya and George

Information Protection Based on Extraction of Square Roots of
Information Protection Based on Extraction of Square Roots of

Families of Shape Functions, Numerical Integration
Families of Shape Functions, Numerical Integration

Summary of lectures.
Summary of lectures.

Slides
Slides

Modular Arithmetic
Modular Arithmetic

Labelled Natural Deduction for Substructural Logics - IME-USP
Labelled Natural Deduction for Substructural Logics - IME-USP

COMMUTATIVE ALGEBRA Contents Introduction 5 0.1. What is
COMMUTATIVE ALGEBRA Contents Introduction 5 0.1. What is

Midterm #3: practice
Midterm #3: practice

On the definition of ulp(x)
On the definition of ulp(x)

Abstract Algebra - UCLA Department of Mathematics
Abstract Algebra - UCLA Department of Mathematics

topology : notes and problems
topology : notes and problems

COMMUTATIVE ALGEBRA Contents Introduction 5
COMMUTATIVE ALGEBRA Contents Introduction 5

as a PDF
as a PDF

Document
Document

9-12 Unit 2: Equations
9-12 Unit 2: Equations

Numbers and Algebra - OSU Department of Mathematics
Numbers and Algebra - OSU Department of Mathematics

LECTURE NOTES IN TOPOLOGICAL GROUPS 1. Lecture 1
LECTURE NOTES IN TOPOLOGICAL GROUPS 1. Lecture 1

Spectra of Small Categories and Infinite Loop Space Machines
Spectra of Small Categories and Infinite Loop Space Machines

Mathematical Olympiads 2000–2001
Mathematical Olympiads 2000–2001

... equality indeed holds for arbitrarily large n. Define (m0 , n0 ) = (1, 1) and (mk+1 , nk+1 ) = (2mk + 3nk , mk + 2nk ) for k ≥ 1. It is easily verified that m2k+1 − 3n2k+1 = m2k − 3n2k . Thus, because the equation 3n2k −2 = m2k holds for k = 0, it holds for all k ≥ 1. Because n1 , n2 , . . . ...
Wilson`s Theorem and Fermat`s Theorem
Wilson`s Theorem and Fermat`s Theorem

... • If n is a positive integer, φ(n) is the number of integers in the range {1, . . . , n} which are relatively prime to n. φ is called the Euler phi-function. • Euler’s theorem generalizes Fermat’s theorem to the case where the modulus is not prime. It says that if n is a positive integer and (a, n) ...
Decomposition numbers for finite Coxeter groups and generalised
Decomposition numbers for finite Coxeter groups and generalised

Mathematical Olympiads 2000–2001
Mathematical Olympiads 2000–2001

... equality indeed holds for arbitrarily large n. Define (m0 , n0 ) = (1, 1) and (mk+1 , nk+1 ) = (2mk + 3nk , mk + 2nk ) for k ≥ 1. It is easily verified that m2k+1 − 3n2k+1 = m2k − 3n2k . Thus, because the equation 3n2k −2 = m2k holds for k = 0, it holds for all k ≥ 1. Because n1 , n2 , . . . ...

Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.