|
Permutable Prime
A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes. In base 10, all the permutable primes with fewer than 49,081 digits are known : 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ... Of the above, there are 16 unique permutation sets, with smallest elements :2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031, ... Note R''n'' = \tfrac is a repunit, a number consisting only of ''n'' ones (in base 10). Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits. All permutable primes of two or more digits are compo ... [...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 alway ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
113 (number)
113 (one hundred ndthirteen) is the natural number following 112 and preceding 114. Mathematics * 113 is the 30th prime number (following 109 and preceding 127), so it can only be divided by one and itself. 113 is a Sophie Germain prime, an emirp, an isolated prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1. 113 is also an Eisenstein prime with no imaginary part and real part of the form 3n - 1. In base 10, this prime is a primeval number, and a permutable prime with 131 and 311. *113 is a highly cototient number and a centered square number. *113 is the denominator of 355/113, an accurate approximation to . See also * 113 (other) * A113 A113 (sometimes A-113, A-1-13, A1-13 or A11-3) is an inside joke and Easter egg in media developed by alumni of California Institute of the Arts, referring to the classroom used by graphic design and character animation students. History Student ... is A Pixar recurring inside jok ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Repdigits are the representation in base B of the number x\frac where 0 1 and ''n'', ''m'' > 2 : **(''p'', ''x'', ''y'', ''m'', ''n'') = (31, 5, 2, 3, 5) corresponding to 31 = 111112 = 1115, and, **(''p'', ''x'', ''y'', ''m'', ''n'') = (8191, 90, 2, 3, 13) corresponding to 8191 = 11111111111112 = 11190, with 11111111111 is the repunit with thirteen digits 1. *For each sequence o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Root Modulo N
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo'' if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base modulo . So is a ''primitive root modulo'' if and only if is a generator of the multiplicative group of integers modulo . Gauss defined primitive roots in Article 57 of the '' Disquisitiones Arithmeticae'' (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime . In fact, the ''Disquisitiones'' contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. Elementary example The number 3 is a primitive root modulo 7 becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duodecimal
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (decimal 14). Similarly, in duodecimal, "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth", respectively). Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses and , as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positional Number System
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the value may be negated if placed before another digit). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string. The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is present today ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort 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 test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...ing. 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repunit Prime
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreations in the Theory of Numbers''. A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of March 2022, the largest known prime number , the largest probable prime ''R''8177207 and the largest elliptic curve primality prime ''R''49081 are all repunits. Definition The base-''b'' repunits are defined as (this ''b'' can be either positive or negative) :R_n^\equiv 1 + b + b^2 + \cdots + b^ = \qquad\mbox, b, \ge2, n\ge1. Thus, the number ''R''''n''(''b'') consists of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n'' = 1 and ''n'' = 2 are :R_1^ 1 \qquad \text \qquad R_2^ b+1\qquad\text\ , b, \ge2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
