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Noncototient
In mathematics, a noncototient is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of coprime integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for Euler's totient function, has no solution for ''m''. The ''cototient'' of ''n'' is defined as ''n'' − φ(''n''), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then :pq - \varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \, It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 1=2-\phi(2), 3 = 9 - \phi(9) and 5 = 25 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
116 (number)
116 (one hundred ndsixteen) is the natural number following 115 and preceding 117. In mathematics 116 is a noncototient, meaning that there is no solution to the equation , where stands for Euler's totient function. 116! + 1 is a factorial prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every ''k'', the union of the first ''k'' subsets is a consecutive sequence of integers. There are 116 different 6×6 Costas arrays.. See also *116 (other) 116 (''one hundred and sixteen'') may refer to: *116 (number) *AD 116 *116 BC *116 (Devon and Cornwall) Engineer Regiment, Royal Engineers, a military unit *116 (MBTA bus) *116 (New Jersey bus) *116 (hip hop group), a Christian hip hop collective * ... References {{DEFAULT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10 (number)
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. It is the first double-digit number. The reason for the choice of ten is assumed to be that humans have ten fingers ( digits). Anthropology Usage and terms * A collection of ten items (most often ten years) is called a decade. * The ordinal adjective is ''decimal''; the distributive adjective is ''denary''. * Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten. * To reduce something by one tenth is to ''decimate''. (In ancient Rome, the killing of one in ten soldiers in a cohort was the punishment for cowardice or mutiny; or, one-tenth of the able-bodied men in a village as a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.) Other * The number of kingdoms in Five Dyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
34 (number)
34 (thirty-four) is the natural number following 33 and preceding 35. In mathematics 34 is the ninth distinct semiprime and has four divisors including one and itself. Its neighbors, 33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers with the same number of divisors as it has. It is the ninth Fibonacci number and a companion Pell number. Since it is an odd-indexed Fibonacci number, 34 is a Markov number, appearing in solutions with other Fibonacci numbers, such as (1, 13, 34), (1, 34, 89), etc. 34 is the magic constant of a 4 by 4 normal magic square: This number is also the magic constant of n-Queens Problem for n = 4. 34 is a heptagonal number. There are 34 topologically distinct convex heptahedra, excluding mirror images. There is no solution to the equation φ(''x'') = 34, making 34 a nontotient. Nor is there a solution to the equation ''x'' − φ(''x'') = 34, making 34 a noncototient. It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
222 (number)
222 (two hundred ndtwenty-two) is the natural number following 221 and preceding 223. In mathematics It is a decimal repdigit and a strobogrammatic number (meaning that it looks the same turned upside down on a calculator display). It is one of the numbers whose digit sum in decimal is the same as it is in binary. 222 is a noncototient, meaning that it cannot be written in the form ''n'' − φ(''n'') where φ is Euler's totient function counting the number of values that are smaller than ''n'' and relatively prime to it. There are exactly 222 distinct ways of assigning a meet and join operation to a set of ten unlabelled elements in order to give them the structure of a lattice, and exactly 222 different six-edge polystick In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
52 (number)
52 (fifty-two) is the natural number following 51 and preceding 53. In mathematics Fifty-two is * the 5th Bell number, the number of ways to partition a set of 5 objects. * a decagonal number. * an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is not equal to ''x'' − φ(''x'') for any ''x''. * a vertically symmetrical number. In science *The atomic number of tellurium Astronomy *Messier object M52, a magnitude 8.0 open cluster in the constellation Cassiopeia, also known as NGC 7654. *The New General Cataloguebr>objectNGC 52, a spiral galaxy in the constellation Pegasus. In other fields Fifty-two is: *The approximate number of weeks in a year. 52 weeks is 364 days, while the tropical year is 365.24 days long. According to ISO 8601, most years have 52 weeks while some have 53. *A significant number in the Maya calendar *On the modern piano, the number of white keys (notes in the C major scale) *The number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
58 (number)
58 (fifty-eight) is the natural number following 57 (number), 57 and preceding 59 (number), 59. In mathematics Fifty-eight is an 11-Polygonal number, gonal number, and a Smith number. Given 58, the Mertens function returns 0 (number), 0. 58 is the smallest integer whose square root has a continued fraction with periodic continued fraction, period 7. 58 is equal to the sum of the first seven consecutive prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 = 58. This is a difference of 1 from the 17th prime number and 7th super-prime, 59 (number), 59. There is no solution to the equation ''x'' – Euler's totient function, φ(''x'') = 58, making 58 a noncototient. However, the sum of the totient function for the first thirteen integers is 58. The regular icosahedron produces 58 distinct stellations. It's dual polyhedron, the regular dodecahedron, only produces three stellations, while the regular octahedron produces a single one, the stella octangula. Neither the tetrahedron nor the cube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
86 (number)
86 (eighty-six) is the natural number following 85 (number), 85 and preceding 87 (number), 87. In mathematics 86 is: * nontotient and a noncototient. * the 25th distinct semiprime and the 13th of the form (2×q). * an Erdős–Woods number, since it is possible to find sequences of 86 consecutive integers such that each inner member shares a factor with either the first or the last member. * a happy number and a self number in base 10. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is the sum of the first two of these). It is conjectured that 86 is the largest n for which the decimal expansion of 2n contains no 0. 86 = (8 × 6 = 48) + (4 × 8 = 32) + (3 × 2 = 6). That is, 86 is equal to the sum of the numbers formed in calculating its Persistence of a number, multiplicative persistence. In science * 86 is the atomic number of radon. * There are 86 metals on the modern periodic table. In other fields *In American English, and particularly in the food ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
100 (number)
100 or one hundred (Roman numeral: C) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. In mathematics 100 is the square of 10 (in scientific notation it is written as 102). The standard SI prefix for a hundred is " hecto-". 100 is the basis of percentages (''per cent'' meaning "per hundred" in Latin), with 100% being a full amount. 100 is a Harshad number in decimal, and also in base-four, a base in-which it is also a self-descriptive number. 100 is the sum of the first nine prime numbers, from 2 through 23. It is also divisible by the number of primes below it, 25. 100 cannot be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. 100 has a reduced totient of 20, and an Euler totient of 40. A totient value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
146 (number)
146 (one hundred ndforty-six) is the natural number following 145 and preceding 147. In mathematics 146 is an octahedral number, the number of spheres that can be packed into in a regular octahedron with six spheres along each edge. For an octahedron with seven spheres along each edge, the number of spheres on the surface of the octahedron is again 146. It is also possible to arrange 146 disks in the plane into an irregular octagon with six disks on each side, making 146 an octo number. There is no integer with exactly 146 coprimes less than it, so 146 is a nontotient. It is also never the difference between an integer and the total of coprimes below it, so it is a noncototient. And it is not the sum of proper divisors of any number, making it an untouchable number. There are 146 connected partially ordered sets with four labeled elements. See also * 146 (other) 146 may refer to: *146 (number), a natural number * AD 146, a year in the 2nd century AD *146 BC, a year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesel Number
In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the form is instead k\times2^n+1, then ''k'' is a Sierpinski number. Riesel Problem In 1956, Hans Riesel showed that there are an infinite number of integers ''k'' such that k\times2^n-1 is not prime for any integer ''n''. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any ''k'' less than 509203, it is conjectured to be the smallest Riesel number. To check if there are ''k'' ''k'') :2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and topology. He published over 700 papers and 50 books. Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet, and the Sierpiński curve), as are Sierpiński numbers and the associated Sierpiński problem. Educational background Sierpiński enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated four years later. In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpiński was awarded a gold medal for his essay, thus laying the foundation for his first major mathematical contribution. Unwilling for his work to be pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
