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Noncototient
In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ..., has no solution for . The '' cototient'' of is defined as , 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 can be represented as a sum of two distinct primes and , then \begin pq - \varphi(pq) &= pq - (p-1)(q-1) \\ &= p + q - 1 \\ &= n - 1. \end It is expected that every even number larger than 6 is a sum of two disti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
100 (number)
100 or one hundred (Roman numerals, Roman numeral: C) is the natural number following 99 (number), 99 and preceding 101 (number), 101. In mathematics 100 is the square of 10 (number), 10 (in scientific notation it is written as 102). The standard SI prefix for a hundred is "Hecto-, hecto-". 100 is the basis of percentages ( meaning "by the 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 (number), 23. It is also divisible by the number of primes below it, 25 (number), 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 Carmichael function, reduced totient of 20, and an Euler totient of 40. A totient value of 100 is obtained from four numbers: 101 (number), 101, 125 (number), 125, 202 (number), 202, and 250 (number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 In number theory, 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 equiv ... 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
10 (number)
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language. Name The number "ten" originates from the Proto-Germanic root "*tehun", which in turn comes from the Proto-Indo-European root "*dekm-", meaning "ten". This root is the source of similar words for "ten" in many other Germanic languages, like Dutch, German, and Swedish. The use of "ten" in the decimal system is likely due to the fact that humans have ten fingers and ten toes, which people may have used to count by. Linguistics * 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
34 (number)
34 (thirty-four) is the natural number following 33 (number), 33 and preceding 35 (number), 35. In mathematics 34 is the twelfth semiprime, with four divisors including 1 and itself. Specifically, 34 is the ninth distinct semiprime, it being the sixth of the form 2 \times q. Its neighbors 33 (number), 33 and 35 (number), 35 are also distinct semiprimes with four divisors each, where 34 is the smallest number to be surrounded by numbers with the same number of divisors it has. This is the first distinct semiprime treble cluster, the next being (85 (number), 85, 86 (number), 86, 87 (number), 87). 34 is the sum of the first two perfect numbers 6 + 28 (number), 28, whose difference is its composite number, composite index (22 (number), 22). Its reduced totient and Euler totient values are both 16 (or 42 = 24). The sum of all its divisors aside from one equals 53 (number), 53, which is the sixteenth prime number. There is no solution to the equation Euler's totient function, φ(''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
52 (number)
52 (fifty-two) is the natural number following 51 (number), 51 and preceding 53 (number), 53. In mathematics Fifty-two is * a composite number; a square-prime, of the form where is some prime larger than It is the sixth of this form and the fifth of the form * the 5th Bell number, the number of ways to partition a set of 5 objects. * a decagonal number. * with an aliquot sum of 46 (number), 46; within an aliquot sequence of seven composite numbers to the prime in the 3 (number), 3-aliquot tree. This sequence does not extend above 52 because it is, * an untouchable number, since it is never the sum of proper divisors of any number. It is the first untouchable number larger than 2 and 5. * a noncototient since it is not equal to for any * a vertically symmetrical number. In other fields Fifty-two is: * The number of cards in a standard deck of playing cards, not counting joker (playing card), Jokers or advertisement cards References {{Integers, zero In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
58 (number)
58 (fifty-eight) is the natural number following 57 and preceding 59. In mathematics 58 is a composite number with four factors: 1, 2, 29, and 58. Other than 1 and the number itself, 58 can be formed by multiplying two primes 2 and 29, making it a semiprime. 58 is not divisible by any square number other than 1, making it a square-free integer A semiprime that is not square numbers is called a squarefree semiprime, and 58 is among them. 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 seventeenth prime number and seventh super-prime, 59. 58 has an aliquot sum of 32 within an aliquot sequence of two composite numbers (58, 32, 31, 1, 0) in the 31-aliquot tree. There is no solution to the equation x - \varphi(x) = 58, making fifty-eight the sixth noncototient; however, the totient summatory function over the first thirteen integers is 58. On the other hand, the Euler totient of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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). * together with 85 (number) , 85 and 87 (number), 87, forms the middle semiprime in the 2nd cluster of three consecutive semiprimes; the first comprising 33 (number), 33, 34 (number), 34, 35 (number), 35. * 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. * with an aliquot sum of 46 (number), 46; itself a semiprime, within an aliquot sequence of seven members (86,46 (number), 46,26 (number), 26,16 (number), 16,15 (number), 15,9 (number), 9,4 (number), 4,3 (number), 3,1 (number), 1,0) in the Prime 3 (number), 3-aliquot tree. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
116 (number)
116 (one hundred [and] sixteen) is the natural number following 115 (number), 115 and preceding 117 (number), 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) References {{DEFAULTSORT:116 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
134 (number)
134 (one hundred ndthirty-four) is the natural number following 133 and preceding 135. In mathematics 134 is a nontotient since there is no integer with exactly 134 coprimes below it. And it is a noncototient since there is no integer with 134 integers with common factors below it. 134 is _8C_1 + _8C_3 + _8C_4. In Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...s, 134 is a Friedman number since CXXXIV = XV * (XC/X) - I. {{DEFAULTSORT:134 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 Sierpiński 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. Early life and education Sierpiński was born in 1882 in Warsaw, Congress Poland, to a doctor father Konstanty and mother Ludwika (''née'' Łapińska). His abilities in mathematics were evident from childhood. He enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated five 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 Vorono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Paul Erdős
Paul Erdős ( ; 26March 191320September 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 on 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 was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
