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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), 3 (2! + 1), 5 (3! − 1), 7 (3! + 1), 23 (4! − 1), 719 (6! − 1), 5039 (7! − 1), 39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1), ... ''n''! − 1 is prime for : :''n'' = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003, ... (resulting in 27 factorial primes) ''n''! + 1 is prime for : :''n'' = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429, ... (resulting in 24 factorial primes - the prime 2 is repeated) ... [...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] |
Divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PrimeGrid
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Pages
The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms. , the 5,000th prime has around 412,000 digits.. Retrieved on 2018-02-12. The PrimePages has articles on primes and primality testing. It includes "The Prime Glossary" with articles on hundreds of glosses related to primes, and "Prime Curios!" with thousands of curios about specific numbers. The database started as a list of titanic primes (primes with at least 1000 decimal digits) by Samuel Yates. In subsequent years, the whole top-5,000 has consisted of gigantic primes (primes with at least 10,000 decimal digits). Primes of special forms are kept on the current lists if they are titanic and in the top-20 or top-5 for their form. See also *List of prime numbers This is a list of articles about pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial Prime
In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that : ''pn''# − 1 is prime for ''n'' = 2, 3, 5, 6, 13, 24, ... : ''pn''# + 1 is prime for ''n'' = 0, 1, 2, 3, 4, 5, 11, ... The first term of the second sequence is 0 because ''p''0# = 1 is the empty product, and thus ''p''0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because ''p''1# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are : 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 , the largest known primorial prime (of the form ''p''''n''# − 1) is 3267113# − 1 (''n'' = 234,725) with 1,418,398 digits, found by the PrimeGrid project. , the largest known prime of the form ''p''''n''# + 1 is 392113# + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g''''n'' or ''g''(''p''''n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ We have ''g''1 = 1, ''g''2 = ''g''3 = 2, and ''g''4 = 4. The sequence (''g''''n'') of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... . By the definition of ''g''''n'' every prime can be written as :p_ = 2 + \sum_^n g_i. Simple observations The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps ' ... [...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] |
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] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
23 (number)
23 (twenty-three) is the natural number following 22 and preceding 24. In mathematics Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 as well as 29. Twenty-three is also the fifth factorial prime, and the second Woodall prime. It is an Eisenstein prime with no imaginary part and real part of the form 3''n'' − 1. 23 is the fifth Sophie Germain prime and the fourth safe prime, 23 is the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23 but 23 is not one more than a multiple of 14, 23 is a Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to ''x'' − φ(''x'') for the integers 95, 119, 143, 529. It is also a happy number in base-10. *In decimal, 23 is the second Smarandache–Wellin prime, as it is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7 (number)
7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven Classical planets resulted in seven being the number of days in a week. It is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky. It is the first natural number whose pronunciation contains more than one syllable. Evolution of the Arabic digit In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase vertically inverted. The western Ghubar Arabs' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arabs developed the digit fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
