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Wilson Quotient
The Wilson quotient ''W''(''p'') is defined as: :W(p) = \frac If ''p'' is a prime number, the quotient is an integer by Wilson's theorem; moreover, if ''p'' is composite, the quotient is not an integer. If ''p'' divides ''W''(''p''), it is called a Wilson prime. The integer values of ''W''(''p'') are : : ''W''(2) = 1 : ''W''(3) = 1 : ''W''(5) = 5 : ''W''(7) = 103 : ''W''(11) = 329891 : ''W''(13) = 36846277 : ''W''(17) = 1230752346353 : ''W''(19) = 336967037143579 : ... It is known that :W(p)\equiv B_-B_\pmod, :p-1+ptW(p)\equiv pB_\pmod{p^2}, where B_k is the ''k''-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting t=1 and t=2. See also * Fermat quotient In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as= 3/ref> The smallest solutions of ''q'p''(''a'') ≡ 0 (mod ''p'') with ' ... [...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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson's Theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of modular arithmetic), the factorial (n - 1)! = 1 \times 2 \times 3 \times \cdots \times (n - 1) satisfies :(n-1)!\ \equiv\; -1 \pmod n exactly when ''n'' is a prime number. In other words, any number ''n'' is a prime number if, and only if, (''n'' − 1)! + 1 is divisible by ''n''. History This theorem was stated by Ibn al-Haytham (c. 1000 AD), and, in the 18th century, by John Wilson. Edward Waring announced the theorem in 1770, although neither he nor his student Wilson could prove it. Lagrange gave the first proof in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but he never published it. Example For each of the values of ''n'' from 2 to 30, the following table shows the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson Prime
In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham. The only known Wilson primes are 5, 13, and 563 . Costa et al. Write that "the case p=5 is trivial", and credit the observation that 13 is a Wilson prime to . Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem. If any others exist, they must be greater than 2 × 1013. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval ,y/math> is about \log\log_x y. Several compute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and indepe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Quotient
In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as= 3/ref> The smallest solutions of ''q''''p''(''a'') ≡ 0 (mod ''p'') with ''a'' = ''n'' are: :2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... {{OEIS, id=A039951 A pair (''p'', ''r'') of prime numbers such that ''q''''p''(''r'') ≡ 0 (mod ''p'') and ''q''''r''(''p'') ≡ 0 (mod ''r'') is called a Wieferich pair. References External links * Gottfried HelmsFermat-/Euler-quotients (''a''''p''-1 – 1)/''p''''k'' with arbitrary ''k'' * Richard FischerFermat quotients B^(P-1) 1 (mod P^2) Number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
