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Sublime Number
In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number. The number 12, for example, is a sublime number. It has a perfect number of positive factors ( 6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28. There are only two known sublime numbers: 12 and (2126)(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1) .Clifford A. Pickover Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ..., ''Wonders of Numbers, Adventures in Mathematics, Mind and Meaning'' New York: Oxford University Press (2003): 215 The second of these has 76 decimal digits: :6,086,555,670,238,378,989,670,371,734,243,169,622,657 ... [...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 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 analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p-1 for positive integer p—what is now called a Mersenne prime. Two millen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind. For example, is a factorization of the integer , and is a factorization of the polynomial . Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any x can be trivially written as (xy)\times(1/y) whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be furthe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
12 (number)
12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by 2, 3, 4, and 6. It is the number of years required for an orbital period of Jupiter. It is central to many systems of timekeeping, including the Western calendar and units of time of day and frequently appears in the world's major religions. Name Twelve is the largest number with a single-syllable name in English. Early Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of eleven and twelve, the former use of "hundred" to refer to groups of 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would normally understand them that way. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance. Derived from Old English, ' and ' are first attested in the 10th-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6 (number)
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. In mathematics Six is the smallest positive integer which is neither a square number nor a prime number; it is the second smallest composite number, behind 4; its proper divisors are , and . Since 6 equals the sum of its proper divisors, it is a perfect number; 6 is the smallest of the perfect numbers. It is also the smallest Granville number, or \mathcal-perfect number. As a perfect number: *6 is related to the Mersenne prime 3, since . (The next perfect number is 28.) *6 is the only even perfect number that is not the sum of successive odd cubes. *6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, . Six is the only number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". Six is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
28 (number)
28 (twenty-eight) is the natural number following 27 and preceding 29. In mathematics It is a composite number, its proper divisors being 1, 2, 4, 7, and 14. Twenty-eight is the second perfect number - it is the sum of its proper divisors: 1+2+4+7+14=28. As a perfect number, it is related to the Mersenne prime 7, since 2^\times (2^-1)=28. The next perfect number is 496, the previous being 6. Twenty-eight is the sum of the totient function for the first nine integers. Since the greatest prime factor of 28^+1=785 is 157, which is more than 28 twice, 28 is a Størmer number. Twenty-eight is a harmonic divisor number, a happy number, a triangular number, a hexagonal number, a Leyland number of the second kind and a centered nonagonal number. It appears in the Padovan sequence, preceded by the terms 12, 16, 21 (it is the sum of the first two of these). It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its decimal digits: 2, 8, 10, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford A
Clifford may refer to: People * Clifford (name), an English given name and surname, includes a list of people with that name * William Kingdon Clifford *Baron Clifford * Baron Clifford of Chudleigh *Baron de Clifford * Clifford baronets *Clifford family (bankers) *Jaryd Clifford *Justice Clifford (other) *Lord Clifford (other) Arts, entertainment, and media *''Clifford the Big Red Dog'', a series of children's books ** Clifford (character), the central character of ''Clifford the Big Red Dog'' ** ''Clifford the Big Red Dog'' (2000 TV series), 2000 animated TV series **''Clifford's Puppy Days'', 2003 animated TV series **''Clifford's Really Big Movie'', 2004 animated movie ** ''Clifford the Big Red Dog'' (2019 TV series), 2019 animated TV series ** ''Clifford the Big Red Dog'' (film), 2021 live-action movie * ''Clifford'' (film), a 1994 film directed by Paul Flaherty *Clifford (Muppet) Mathematics * Clifford algebra, a type of associative algebra, named after Will ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor Function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. Definition The sum of positive divisors function σ''z''(''n''), for a real or complex number ''z'', is defined as the sum of the ''z''th powers of the positive divisors of ''n''. It can be expressed in sigma notation as :\sigma_z(n)=\sum_ d^z\,\! , where is shorthand for "''d' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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