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Giuga Number
A Giuga number is a composite number ''n'' such that for each of its distinct prime factors ''p''''i'' we have p_i , \left( - 1\right), or equivalently such that for each of its distinct prime factors ''p''''i'' we have p_i^2 , (n - p_i). The Giuga numbers are named after the mathematician Giuseppe Giuga, and relate to his conjecture on primality. Definitions Alternative definition for a Giuga number due to Takashi Agoh is: a composite number ''n'' is a Giuga number if and only if the congruence :nB_ \equiv -1 \pmod n holds true, where ''B'' is a Bernoulli number and \varphi(n) is Euler's totient function. An equivalent formulation due to Giuseppe Giuga is: a composite number ''n'' is a Giuga number if and only if the congruence :\sum_^ i^ \equiv -1 \pmod n and if and only if :\sum_ \frac - \prod_ \frac \in \mathbb. All known Giuga numbers ''n'' in fact satisfy the stronger condition :\sum_ \frac - \prod_ \frac = 1. Examples The sequence of Giuga numbers begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuseppe Giuga
Giuseppe is the Italian form of the given name Joseph, from Latin Iōsēphus from Ancient Greek Ἰωσήφ (Iōsḗph), from Hebrew יוסף. It is the most common name in Italy and is unique (97%) to it. The feminine form of the name is Giuseppina. People with the given name Artists and musicians * Giuseppe Aldrovandini (1671–1707), Italian composer * Giuseppe Arcimboldo (1526 or 1527–1593), Italian painter * Giuseppe Belli (singer) (1732–1760), Italian castrato singer * Giuseppe Gioachino Belli (1791–1863), Italian poet * Giuseppe Castiglione (1829–1908) (1829–1908), Italian painter * Giuseppe Giordani (1751–1798), Italian composer, mainly of opera * Giuseppe Ottaviani (born 1978), Italian musician and disc jockey * Giuseppe Psaila (1891–1960), Maltese Art Nouveau architect * Giuseppe Sammartini (1695–1750), Italian composer and oboist * Giuseppe Sanmartino or Sammartino (1720–1793), Italian sculptor * Giuseppe Santomaso (1907–1990), Italian painter * Giu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takashi Agoh
is a masculine Japanese given name. Possible writings The name Takashi can have multiple different meanings depending on which kanji is used to write it. Some possible writings of the name include: *江詩 - "estuary , inlet, poem" *隆 - "prosperous noble" *喬士 - "high, boasting, samurai, gentleman" *峻 - "high, steep" *崇史 - "adore, revere, chronicler, history" *孝 - "filial piety, serve parents" *節 - "moral courage, integrity" *傑 - "hero, outstanding" Takashi can also be written in hiragana and/or katakana: *タカシ (katakana) *たかし (hiragana) People with the name *Takashi Abe (阿部 隆, born 1967), Japanese shogi player *, Japanese rugby union player *Takashi Amano (天野尚, 1954–2015), Japanese photographer, aquarist and designer *Takashi Aonishi (青西 高嗣), Japanese music artist *Takashi Asahina (朝比奈 隆, 1908–2001), Japanese conductor *, Japanese volleyball player *Takashi Fujii (藤井隆, born 1972), Japanese singer and comedian *Taka ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and indepe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-free Integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes. Examples and variations The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are the case k=2 of the k-almost primes, numbers with exactly k prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are: Formula for number of semiprimes A semiprime counting formula was discovered by E. Noel and G. Panos in 2005. Let \pi_2(n) denote the number of semiprimes less than or equal to n. Then \pi_2(n) = \sum_^ pi(n/p_k) - k + 1 /math> where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Derivative
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. There are many versions of "arithmetic derivatives", including the one discussed in this article (the Lagarias arithmetic derivative), such as Ihara's arithmetic derivative and Buium's arithmetic derivatives. Early history The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911. The arithmetic derivative also appeared in the 1950 Putnam Competition. Definition For natural numbers , the arithmetic derivative In this article we use Oliver Heaviside's notation for the arithmetic derivative of . There are various other notations possible, such as ; a full discussion is available here for general differential operators, of which the arithmetic derivative can be considered one. Heaviside's notation is used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
