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Agoh–Giuga Conjecture
In number theory the Agoh–Giuga conjecture on the Bernoulli numbers ''B''''k'' postulates that ''p'' is a prime number if and only if :pB_ \equiv -1 \pmod p. It is named after Takashi Agoh and Giuseppe Giuga. Equivalent formulation The conjecture as stated above is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that ''p'' is prime if and only if :1^+2^+ \cdots +(p-1)^ \equiv -1 \pmod p which may also be written as :\sum_^ i^ \equiv -1 \pmod p. It is trivial to show that ''p'' being prime is sufficient for the second equivalence to hold, since if ''p'' is prime, Fermat's little theorem states that :a^ \equiv 1 \pmod p for a = 1,2,\dots,p-1, and the equivalence follows, since p-1 \equiv -1 \pmod p. Status The statement is still a conjecture since it has not yet been proven that if a number ''n'' is not prime (that is, ''n'' is composite), then the formula does not hold. It has been shown that a composite number ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...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] |
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] |
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] |
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] |
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] |
Fermat's Little Theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If is not divisible by , that is if is coprime to , Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols: : a^ \equiv 1 \pmod p. For example, if = 2 and = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.. History Pierre de Fermat first stated the theorem in a letter dated October ... [...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] |
Carmichael Number
In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation: :b^n\equiv b\pmod for all integers b. The relation may also be expressed in the form: :b^\equiv 1\pmod. for all integers b which are relatively prime to n. Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 ( Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short). They are infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. Overview Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''b'', the number ''b'' − ''b'' is an integer multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Wilson's Theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of modular arithmetic), the factorial (n - 1)! = 1 \times 2 \times 3 \times \cdots \times (n - 1) satisfies :(n-1)!\ \equiv\; -1 \pmod n exactly when ''n'' is a prime number. In other words, any number ''n'' is a prime number if, and only if, (''n'' − 1)! + 1 is divisible by ''n''. History This theorem was stated by Ibn al-Haytham (c. 1000 AD), and, in the 18th century, by John Wilson. Edward Waring announced the theorem in 1770, although neither he nor his student Wilson could prove it. Lagrange gave the first proof in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but he never published it. Example For each of the values of ''n'' from 2 to 30, the following table shows the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manuscripta Mathematica
The Knights of Columbus Vatican Film Library in St. Louis, Missouri is the only collection, outside the Vatican itself, of microfilms of more than 37,000 works from the ''Biblioteca Apostolica Vaticana'', the Vatican Library in Europe. It is located in the Pius XII Memorial Library on the campus of Saint Louis University. History The Library was created by Lowrie J. Daly (1914–2000), with funding from the Knights of Columbus. The goal was to make Vatican and other documents more available to researchers in North America. Microfilming of Vatican manuscripts began in 1951, and according to the Library's website, was the largest microfilming project that had been undertaken up to that date. From 1951 to 1957, twelve million manuscript pages were recorded, from 30,000 different works. This represents approximately 75% of the manuscripts available in the targeted language groups. Other microfilm projects in the 1950s included Jesuit archival material from Rome, archives in both Nort ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
