Unit (ring theory)In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R^× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).
Multiplicative group of integers modulo nIn modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n.
Ramanujan's sumIn number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula where (a, q) = 1 means that a only takes on values coprime to q. Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes.
Order (group theory)In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a.
Fermat's little theoremFermat's little theorem states that if p is a prime number, then for any integer a, the number is an integer multiple of p. In the notation of modular arithmetic, this is expressed as For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is if a is coprime to p, Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7.
Fermat numberIn mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 ; heuristics suggest that there are no more.
Lambert seriesIn mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resumed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1: This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series.
Euler's constantEuler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log: Here, ⌊ ⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43).
PrimorialIn mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors. For the nth prime number pn, the primorial pn# is defined as the product of the first n primes: where pk is the kth prime number.
Prime number theoremIn mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).
