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Sphenic Number
In number theory, a sphenic number (from , 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. Definition A sphenic number is a product ''pqr'' where ''p'', ''q'', and ''r'' are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3- almost primes. Examples The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are : 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... The largest known sphenic number at any time can be obtained by multiplying together the three largest known primes. Divisors All sphenic numbers have exactly eight divisors. If we express the sphenic number as n = p \cdot q \cdot r, where ''p'', ''q'', and ''r'' are distinct primes, then the set of divisors of ''n'' will be: :\left\. The converse does not hold. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
110 (number)
110 (one hundred ndten) is the natural number following 109 and preceding 111. In mathematics 110 is a sphenic number and a pronic number. Following the prime quadruplet (101, 103, 107, 109), at 110, the Mertens function reaches a low of −5. 110 is the sum of three consecutive squares, 110 = 5^2 + 6^2 + 7^2. RSA-110 is one of the RSA numbers, large semiprimes that are part of the RSA Factoring Challenge. In base 10, the number 110 is a Harshad number and a self number. In other fields 110 is also: * 1-1-0, the emergency telephone number used to reach police services in Iran, Germany, Estonia, China, Indonesia, and Japan. Also used to reach the fire and rescue services in Norway and Turkey. * A percentage in the expression "To give 110%", meaning to give a little more effort than one's maximum effort * Lowest number to not be considered a favourite by anyone among 44,000 people surveyed in a 2014 online poll and subsequently adopted by British television show '' QI ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Prime
In number theory, a natural number is called -almost prime if it has prime factors. More formally, a number is -almost prime if and only if , where is the total number of primes in the prime factorization of (can be also seen as the sum of all the primes' exponents): :\Omega(n) := \sum a_i \qquad\mbox\qquad n = \prod p_i^. A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of -almost primes is usually denoted by . The smallest -almost prime is . The first few -almost primes are: : The number of positive integers less than or equal to with exactly prime divisors (not necessarily distinct) is asymptotic to: : \pi_k(n) \sim \left( \frac \right) \frac, a result of Landau Landau (), officially Landau in der Pfalz (, ), is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes, since they include two primes, or second numbers, by analogy with how "prime" means "first". Alternatively non-prime semiprimes are called almost-prime numbers, specifically the "2-almost-prime" biprime and "3-almost-prime" triprime Examples and variations The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are the case k=2 of the k- almost primes, numbers with exactly k prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Polynomials
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n)—and more precisely, because of the failure of unique factorization in their rings of integers—that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n \geq 1, let :\zeta_n=e^\in\C. This is a primitive nth root of unity. Then the nth cyclotomic field is the field extension \mathbb(\zeta_n) of \mathbb generated by \zeta_n. Properties * The nth cyclotomic polynomial :: \Phi_n(x) = \prod_\stackrel\!\!\! \left(x-e^\right) = \prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of \zeta_n over \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted \mu(x). Definition The Möbius function is defined by :\mu(n) = \begin 1 & \text n = 1 \\ (-1)^k & \text n \text k \text \\ 0 & \text n \text > 1 \end The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where \delta_ is the Kronecker delta, \lambda(n) is the Liouville function, Prime omega function, \omega(n) is the number of distinct prime divisors of n, and Prime omega function, \Omega(n) is the number of prime factors of n, counted with multiplicity. Another characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Largest Known Prime
The largest known prime number is , a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a 36-year-old researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS). A prime number is a natural number greater than 1 with no divisors other than 1 and itself. Euclid's theorem proves that for any given prime number, there will always be a higher one, and thus there are infinitely many; there is no largest prime. Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilize a specialized primality test that is faster than the general one. , the seven largest known primes are Mersenne primes. The last eighteen record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2''k'' − 1 is simply ''k'' ones. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
165 (number)
165 (one hundred ndsixty-five) is the natural number following 164 and preceding 166. In mathematics 165 is: *an odd number, a composite number, and a deficient number. *a sphenic number. *a tetrahedral number. *the number of prime knots with 10 crossings. *the sum of the sums of the divisors of the first 14 positive integers. *a self number in base 10. *a palindromic number A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ... in binary (101001012) and bases 14 (BB14), 32 (5532) and 54 (3354). *a unique period in base 2. References External links Number Facts and Trivia: 165The Number 165The Positive Integer 165 Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
154 (number)
154 (one hundred ndfifty-four) is the natural number following 153 and preceding 155. In mathematics 154 is a nonagonal number. Its factorization makes 154 a sphenic number. There is no integer with exactly 154 coprimes below it, making 154 a noncototient, nor is there, in base 10, any integer that added up to its own digits yields 154, making 154 a self number 154 is the sum of the first six factorials, if one starts with 0! and assumes that 0!=1. With just 17 cuts, a pancake can be cut up into 154 pieces ( Lazy caterer's sequence). The distinct prime factors of 154 add up to 20, and so do the ones of 153, hence the two form a Ruth-Aaron pair. 154! + 1 is a factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! + 1 or 1! + 1) .... References * Wells, D. '' The Penguin Diction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
138 (number)
138 (one hundred [and] thirty-eight) is the natural number following 137 (number), 137 and preceding 139 (number), 139. Mathematics 138 is a sphenic number, an Ulam number, an abundant number, and a square-free congruent number. References [Baidu] |
130 (number)
130 (one hundred ndthirty) is the natural number following 129 and preceding 131. In mathematics 130 is a sphenic number. It is a 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 ... since there is no answer to the equation ''x'' - φ(''x'') = 130. 130 is the only integer that is the sum of the squares of its first four divisors, including 1: 12 + 22 + 52 + 102 = 130. 130 is the largest number that cannot be written as the sum of four hexagonal numbers. 130 equals both 27 + 2 and 53 + 5 and is therefore a ''doubly strictly '' number. In other fields * A 130-30 fund or a ratio up to 150/50 is a type of collective investment vehicle References {{DEFAULTSORT:130 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
114 (number)
114 (one hundred [and] fourteen) is the natural number following 113 (number), 113 and preceding 115 (number), 115. In mathematics *114 is an abundant number, a sphenic number and a Harshad number. It is the sum of the first four hyperfactorials, including H(0). At 114, the Mertens function sets a new low of -6, a record that stands until 197. *114 is the smallest positive integer* which has yet to be represented as a3 + b3 + c3, Sums of three cubes, where a, b, and c are integers. It is conjectured that 114 can be represented this way. (*Excluding integers of the form 9k ± 4, for which solutions are known not to exist.) *There is no answer to the equation Euler's totient function, φ(x) = 114, making 114 a nontotient. *114 appears in the Padovan sequence, preceded by the terms 49, 65, 86 (it is the sum of the first two of these). *114 is a repdigit in base 7 (222). See also * 114 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
