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Descartes Number
In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the number would be an odd perfect number if only were a prime number, since the sum-of-divisors function for would satisfy, if 22021 were prime, :\begin \sigma(D) &= (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^2+13+1)\cdot(22021+1) = (13)\cdot(3\cdot19)\cdot(7\cdot19)\cdot(3\cdot61)\cdot(22\cdot1001) \\ &= 3^2\cdot7\cdot13\cdot19^2\cdot61\cdot(22\cdot7\cdot11\cdot13) = 2 \cdot (3^2\cdot7^2\cdot11^2\cdot13^2) \cdot (19^2\cdot61) = 2 \cdot (3^2\cdot7^2\cdot11^2\cdot13^2) \cdot 22021 = 2D, \end where we ignore the fact that 22021 is composite (). A Descartes number is defined as an odd number where and are coprime and , whence is taken as a 'spoof' prime. The example given is the only one currently known. If is an odd almost perfect number,Currently, the only known almost ... [...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] |
Powers Of 2
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the '' American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to records on JSTOR. Mission and Vision The mission of the MAA is to advance the understanding of mathematics and its impact on our world. We envision a society that values the power and beauty of mathematics and fully realizes its potential to promote human flourishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Nicolas Number
In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors. That is, a number is Erdős–Nicolas number when there exists another number such that : \sum_d=n. The first ten Erdős–Nicolas numbers are : 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800 and 91963648. () They are named after Paul Erdős and Jean-Louis Nicolas, who wrote about them in 1975. See also *Descartes number In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the number would be an odd perfect number if onl ..., another type of almost-perfect numbers References Integer sequences {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brigham Young University
Brigham Young University (BYU, sometimes referred to colloquially as The Y) is a private research university in Provo, Utah. It was founded in 1875 by religious leader Brigham Young and is sponsored by the Church of Jesus Christ of Latter-day Saints (LDS Church). BYU offers a variety of academic programs including those in the liberal arts, engineering, agriculture, management, physical and mathematical sciences, nursing, and law. It has 186 undergraduate majors, 64 master's programs, and 26 doctoral programs. It is broadly organized into 11 colleges or schools at its main Provo campus, with some colleges and divisions defining their own admission standards. The university also administers two satellite campuses, one in Jerusalem and one in Salt Lake City, while its parent organization the Church Educational System (CES) sponsors sister schools in Hawaii and Idaho. The university is accredited by the Northwest Commission on Colleges and Universities. Almost all BYU students ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Voight (mathematician)
John Voight may refer to: *Jon Voight (born 1938), American actor * John Voight (athlete) (1926–1993), American sprinter *Wolfgang Zilzer Wolfgang Zilzer (January 20, 1901 – June 26, 1991) was a German-American stage and film actor, often under the stage name Paul Andor. Biography Zilzer was born in Cincinnati, Ohio, to German-Jewish emigrant Max Zilzer, who was employed at the ... (1901–1991), German-American actor who used the stage name "John Voight" * John Voight (mathematician), University of Vermont mathematician and winner of the 2010 Selfridge Prize {{hndis, Voight, John ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube-free
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. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Perfect Number
In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ''n'' such that the sum of all divisors of ''n'' (the sum-of-divisors function ''σ''(''n'')) is equal to 2''n'' − 1, the sum of all proper divisors of ''n'', ''s''(''n'') = ''σ''(''n'') − ''n'', then being equal to ''n'' − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents . Therefore the only known odd almost perfect number is 20 = 1, and the only known even almost perfect numbers are those of the form 2''k'' for some positive number ''k''; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least six prime factors. If ''m'' is an odd almost perfect number then is a Descartes number. Moreover if ''a'' and ''b'' are positive odd integers such that b+3 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and 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 does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
