Modular multiplicative inverseIn mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integera is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as ax≡1(modm),{\displaystyle ax\equiv 1{\pmod {m}},}which is the shorthand way of writing the statement that m divides (evenly) the quantity ax − 1, or, put another way, the remainder after dividing ax by the integer m is 1.
Coprime integersIn number theory, two integersa and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ais prime tob or ais coprime withb.
Equivalence relationY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
ArithmeticBranch of elementary mathematicsThe main arithmetic operations are addition, subtraction, multiplication, and division.Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on.
Number theoryBranch of pure mathematicsThe distribution of prime numbers, a central point of study in number theory, illustrated by an Ulam spiral. It shows the conditional independence between being prime and being a value of certain quadratic polynomials.Number theory is a branch of mathematics devoted primarily to the study of the integers and arithmetic functions.
Euler's totient functionNumber of integers coprime to and less than nNot to be confused with Euler function. without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x){\displaystyle \ln(x)} or loge(x){\displaystyle \log _{e}(x)}.The first thousand values of φ(n).
PolynomialType of mathematical expressionIn mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x{\displaystyle x} is x2−4x+7{\displaystyle x^{2}-4x+7}.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
IntegerAn integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.
Disquisitiones ArithmeticaeDisquisitiones Arithmeticae (Latin for Arithmetical Investigations) is a textbook on number theory written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24. It had a revolutionary impact on number theory by making the field truly rigorous and systematic and paved the path for modern number theory.
Carl Friedrich GaussGerman polymath and scholar (1777–1855)Johann Carl Friedrich Gauss ( ; German: Gauß; 30 April 1777 – 23 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. His mathematical contributions spanned the branches of number theory, algebra, analysis, geometry, statistics, and probability.
Clock faceA clock face is the part of an analog clock (or watch) that displays time through the use of a flat dial with reference marks, and revolving pointers turning on concentric shafts at the center, called hands (or arms). In its most basic, globally recognized form, the periphery of the dial is numbered 1 through 12 indicating the hours in a 12-hour cycle, and a short hour hand makes two revolutions in a day. A long minute hand makes one revolution every hour.*As an Amazon Associate I earn from qualifying purchases.