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Harmonic Prime
A harmonic prime is a prime number that divides the numerators of exactly three harmonic numbers In mathematics, the -th harmonic number is the sum of the Multiplicative inverse, reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \f .... Specifically, a harmonic prime is always a factor of the numerators of the partial harmonic sums at positions , , and . For example, the numerators of the fractions given by \sum_^ \frac , \sum_^ \frac , and \sum_^ \frac are 25, 55835135, and 1347822955, each of which is divisible by 5. All prime numbers greater than 5 can also be found at those three indices, but many also appear at other indices. It is conjectured that there are infinitely many harmonic primes. References {{Prime number classes Classes of prime numbers ... [...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 (mathematics), 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 factorization, 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 primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Experimental Mathematics (journal)
''Experimental Mathematics'' is a quarterly scientific journal of mathematics published by A K Peters, Ltd. until 2010, now by Taylor & Francis. The journal publishes papers in experimental mathematics, broadly construed. The journal's mission statement describes its scope as follows: "Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses." Its editor-in-chief is Alexander Kasprzyk (University of Nottingham). History ''Experimental Mathematics'' was established in 1992 by David Epstein, Silvio Levy, and Klaus Peters. ''Experimental Mathematics'' was the first mathematical research journal to concentrate on experimental mathematics and to explicitly acknowledge its importance for mathematics as a general research field. The journal's launching was described as "something of a watershed". Indeed, the launching of the journal in 1992 was surrounded by som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CiteSeerX
CiteSeerX (formerly called CiteSeer) is a public search engine and digital library for scientific and academic papers, primarily in the fields of computer and information science. CiteSeer's goal is to improve the dissemination and access of academic and scientific literature. As a non-profit service that can be freely used by anyone, it has been considered part of the open access movement that is attempting to change academic and scientific publishing to allow greater access to scientific literature. CiteSeer freely provided Open Archives Initiative metadata of all indexed documents and links indexed documents when possible to other sources of metadata such as DBLP and the ACM Portal. To promote open data, CiteSeerX shares its data for non-commercial purposes under a Creative Commons license. CiteSeer is considered a predecessor of academic search tools such as Google Scholar and Microsoft Academic Search. CiteSeer-like engines and archives usually only harvest documents f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
