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Highly Totient Number
A highly totient number k is an integer that has more solutions to the equation \phi(x) = k, where \phi is Euler's totient function, than any integer below it. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440 , with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to \phi(x) = k. The totient of a number x, with prime factorization x=\prod_i p_i^, is the product: :\phi(x)=\prod_i (p_i-1)p_i^. Thus, a highly totient number is a number that has more ways of being expressed as a product of this form than does any smaller number. The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
240 (number)
240 (two hundred ndforty) is the natural number following 239 and preceding 241. In mathematics 240 is: *a semiperfect number. *a concatenation of two of its proper divisors. *a highly composite number since it has 20 divisors total (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240), more than any previous number. *a refactorable number or tau number, since it has 20 divisors and 20 divides 240. *a highly totient number, since it has 31 totient answers, more than any previous integer. *a pronic number since it can be expressed as the product of two consecutive integers, 15 and 16. *palindromic in bases 19 (CC19), 23 (AA23), 29 (8829), 39 (6639), 47 (5547) and 59 (4459). *a Harshad number in bases 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15 (and 73 other bases). *the aliquot sum of 120 and 57121. *part of the 12161-aliquot tree. The aliquot sequence starting at 120 is: 120, 240, 504, 1056, 1968, 3240, 7650, 14112, 32571, 27333, 12161, 1, 0. 240 is the sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Cototient Number
In number theory, a branch of mathematics, a highly cototient number is a positive integer k which is above 1 and has more solutions to the equation :x - \phi(x) = k than any other integer below k and above 1. Here, \phi is Euler's totient function. There are infinitely many solutions to the equation for :k = 1 so this value is excluded in the definition. The first few highly cototient numbers are: : 2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, ... Many of the highly cototient numbers are odd. In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 29 modulo 30. The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since integer factori ... [...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] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nontotient
In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotient if there is no integer ''x'' that has exactly ''n'' coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions ''x'' = 1 and ''x'' = 2. The first few even nontotients are : 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, ... Least ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) :1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 69, 0, 47, 0, 65, 0, 0, 0, 53, 0, 81, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Number
__FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. The late mathematician Jean-Pierre Kahane has suggested that Plato must have known about highly composite numbers as he deliberately chose 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. Ramanujan wrote and titled his paper on the subject in 1915. Examples The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate superior highly composite numbers. The divisors of the first 15 highly composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
720 (number)
720 (seven hundred [and] twenty) is the natural number following 700 (number)#710s, 719 and preceding 700 (number)#720s, 721. It is 6! (6 (number), 6 factorial), a composite number with thirty divisors, more than any number below, making it a highly composite number. It is a Harshad number in every base from binary to decimal. 720 is expressible as the product of consecutive integers in two different ways: , and . There are 49 solutions to the equation Euler's totient function, φ(''x'') = 720, more than any integer below it, making 720 a highly totient number. 720 is a 241-polygonal number, gonal number. In other fields 720 is: * A common vertical display resolution for High-definition television, HDTV (see 720p). * 720° is two full rotations; the term "720" refers to a skateboarding trick. * 720° is also the name of a skateboarding video game. * 720 is a dual area codes 303 and 720, area code in the Denver Metro Area along with 303. * 720° is the sum of all the defect ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
144 (number)
144 (one hundred ndforty-four) is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross. In mathematics 144 is the twelfth Fibonacci number, and the largest one to also be a square, as the square of 12 (which is also its index in the Fibonacci sequence), following 89 and preceding 233. 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 has 16 divisors. 144 is divisible by the value of its φ function, which returns 48 in this case. Also, there are 21 solutions to the equation φ(''x'') = 144, more than any integer below 144, making it a highly totient number. 144 = 27 + 84 + 110 + 133, the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture. 144 is in base 10 a sum-product number, as well as the sum of a twin prime pair (71 + 73). The maximum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
72 (number)
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen (i.e., 60 in duodecimal). In mathematics Seventy-two is a pronic number, as it is the product of 8 and 9. 72 is an abundant number, with a total of 12 factors, and a Euler totient of 24. 72 is also a highly totient number, as there are 17 solutions to the equation φ(''x'') = 72, more than any integer below 72. It is equal to the sum the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors. 144, or twice 72, is also highly totient, as is 576, the square of 24. While 17 different integers have a totient value of 72, the sum of Euler's totient function φ(''x'') over the first 15 integers is 72. 72 is also a Harshad number in decimal, as it is divisible by the sum of its digits. *72 is the smallest Achilles number, as it's a powerful number that is not itself a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
48 (number)
48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens. In mathematics Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17- gonal number. 48 is the smallest number with exactly ten divisors. There are 11 solutions to the equation φ(''x'') = 48, namely 65, 104, 105, 112, 130, 140, 144, 156, 168, 180 and 210. This is more than any integer below 48, making 48 a highly totient number. Since the greatest prime factor of 482 + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number. 48 is a Harshad number in base 10. It has 24, 2, 12, and 4 as factors. In science *The atomic number of cadmium. *The number of Ptolemaic constellations. *The number of symmetries of a cube. Astronomy *Messier object M48, a magnitude 5.5 open cluster in the constellation Hydra. *The New General Cataloguebr>objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
