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31st Illinois General Assembly
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number. In mathematics 31 is the 11th prime number. It is a superprime and a self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is a lucky prime and a happy number; two properties it shares with 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its twin prime, 29. 31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge, from combinations of known Fermat primes of the form 22''n'' + 1. 31 is the third Mersenne prime of the form 2''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit signed binary integer in computing. After 3, it is the second Mersenne prime not to be a double Mersenne prime. 127, which is the 31st prime number, i ... [...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] |
2,147,483,647
The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes. The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772. Euler used trial division, improving on Pietro Cataldi's method, so that at most 372 divisions were needed. It thus improved upon the previous record-holding prime, 6,700,417, also discovered by Euler, forty years earlier. The number 2,147,483,647 remained the largest known prime until 1867. In computing, this number is the largest value that a signed 32-bit integer field can hold. Barlow's prediction At the time of its discovery, 2,147,483,647 was the largest known prime number. In 1811, Peter Barlow, not anticipating future interest in perfect numbers, wrote (in ''An Elementary Investigation of the Theory of Numbers''): Euler ascertained that 231 − 1 = 2147483647 is a prime number; and this is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dana S
Dana may refer to: People Given name * Dana (given name) Surname * Dana (surname) * Dana family of Cambridge, Massachusetts ** James Dwight Dana (1813–1895), scientist, zoological author abbreviation Dana Nickname or stage name * Dana International, stage name of singer Sharon Cohen * Dana Shum, the Shaw Brothers Hong Kong actress from 1973 to 1979 * Dana, stage name of Dana Rosemary Scallon (born 1951), Irish singer and former politician * Dana (South Korean singer) (born 1986), South Korean pop singer Places Ancient world * Ancient Dana or Tyana in Cappadocia, capital of a Neo-Hittite kingdom in the 1st millennium BC * Ancient Dana possibly associated with Tynna in Cappadocia Canada * CFS Dana, a former military radar installation in Saskatchewan, Canada * Dana Lake, a lake in Eeyou Istchee Baie-James, Quebec, Canada Ethiopia * Dana, Ethiopia, a village Iran * Dana County, an administrative subdivision of Iran * Dana Rural District, an administrative subdi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner Tree Problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals (but may include additional vertices). Further well-known variants are the ''Euclidean Steiner tree problem'' and the '' rectilinear minimum Steiner tree problem''. The Steiner tree problem in graphs can be seen as a generalization of two other famous combinatorial optimization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Decagonal Number
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by the formula :5n^2+5n+1 \, Thus, the first few centered decagonal numbers are : 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, ... Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can be reckoned by multiplying the (''n'' − 1)th triangular number by ''k'', 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in 1. Another consequence of this relation to triangular numbers is the simple recurrence relation for centered decagonal numbers: :CD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Pentagonal Number
A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered pentagonal number for ''n'' is given by the formula :P_=, n\geq1 The first few centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976 . Properties *The parity of centered pentagonal numbers follows the pattern odd-even-even-odd, and in base 10 the units follow the pattern 1-6-6-1. *Centered pentagonal numbers follow the following Recurrence relations: :P_=P_+5n , P_0=1 :P_=3(P_-P_)+P_ , P_0=1,P_1=6,P_2=16 *Centered pentagonal numbers can be expressed using Triangular Numbers: :P_=5T_+1 See also *Pentagonal number *Polygonal number *Centered polygonal number The centered polygonal numbers are a class of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Triangular Number
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). Properties *The gnomon of the ''n''-th centered triangular number, corresponding to the (''n'' + 1)-th triangular layer, is: ::C_ - C_ = 3(n+1). *The ''n''-th centered triangular number, corresponding to ''n'' layers ''plus'' the center, is given by the formula: ::C_ = 1 + 3 \frac = \frac. *Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number. *Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
496 (number)
496 (four hundred ndninety-six) is the natural number following 495 and preceding 497. In mathematics 496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case. A triangular number and a hexagonal number, 496 is also a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular prime-indexed number is a prime number. It is the largest happy number less than 500. There is no solution to the equation φ(''x'') = 496, making 496 a nontotient. ''E''8 has real dim ... [...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 millennia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
127 (number)
127 (one hundred ndtwenty-seven) is the natural number following 126 and preceding 128. It is also a prime number. In mathematics *As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known mersenne prime exponent for a Mersenne number, 2^-1, which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years. **2^-1 is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime. ** Furthermore, 127 is equal to 2^-1, and 7 is equal to 2^-1, and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime. *There are a total of 127 prime numbers between 2,000 and 3,000. *127 is also a cuban prime of the form p=\frac, x=y+1. The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Mersenne Prime
In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is prime. Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and Lists''at the Prime Pages. : :M_ = M_3 = 7 :M_ = M_7 = 127 :M_ = M_ = 2147483647 :M_ = M_ = 170141183460469231731687303715884105727 Double Mersenne primes A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number ''M''''p'' can be prime only if ''p'' is prime, (see Mersenne prime for a proof), a double Mersenne number M_ can be prime only if ''M''''p'' is itself a Mersenne prime. For the first values of ''p'' for which ''M''''p'' is prime, M_ is known to be prime for ''p'' = 2, 3, 5, 7 while explicit factors of M_ have been found for ''p'' = 13, 17, 19, and 31. Thus, the smallest candidate for the next double Mersenne prime is M_, or 22305843009213693951 − 1. Being approximately 1.695, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
