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Wolstenholme Number
A Wolstenholme number is a number that is the numerator of the generalized harmonic number ''H''''n'',2. The first such numbers are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... . These numbers are named after Joseph Wolstenholme, who proved Wolstenholme's theorem In mathematics, Wolstenholme's theorem states that for a prime number p \geq 5, the congruence : \equiv 1 \pmod holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a multiple ... on modular relations of the generalized harmonic numbers. References * Integer sequences {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...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 comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Wolstenholme
Joseph Wolstenholme (30 September 1829 – 18 November 1891) was an English mathematician. Wolstenholme was born in Eccles near Salford, Lancashire, England, the son of a Methodist minister, Joseph Wolstenholme, and his wife, Elizabeth, ''née'' Clarke. He graduated from St John's College, Cambridge as Third Wrangler in 1850 and was elected a fellow of Christ's College in 1852. Collaborating with Percival Frost, a ''Treatise on Solid Geometry'' was published in 1863. Wolstenholme served as Examiner in 1854, 1856, and 1863 for Cambridge Mathematical Tripos, and according to Andrew Forsyth his book ''Mathematical Problems'' made a significant contribution to mathematical education: :...gathered together from many examination papers to form a volume, which was considerably amplified in later editions, they exercised a very real influence upon successive generations of undergraduates; and "Wolstenholme's Problems" have proved a help and stimulus to many students. In 1869 he resig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolstenholme's Theorem
In mathematics, Wolstenholme's theorem states that for a prime number p \geq 5, the congruence : \equiv 1 \pmod holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo ''p''2, which holds for p \geq 3. An equivalent formulation is the congruence : \equiv \pmod for p \geq 5, which is due to Wilhelm Ljunggren (and, in the special case b = 1, to J. W. L. Glaisher) and is inspired by Lucas' theorem. No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo ''p''4 is called a Wolstenholme prime (see below). As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers: :1+++\dots+ \equiv 0 \pmod \mbox :1+++\dot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
