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Eisenstein Prime
In mathematics, an Eisenstein prime is an Eisenstein integer : z = a + b\,\omega, \quad \text \quad \omega = e^, that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units , itself and its associates. The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime. Characterization An Eisenstein integer is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold: # is equal to the product of a unit and a natural prime of the form (necessarily congruent to ), # is a natural prime (necessarily congruent to 0 or ). It follows that the square of the absolute value of every Eisenstein prime is a natural prime or the square of a natural prime. In base 12 (written with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , ): The natural Eisenstein primes are exactly the natural primes ending with 5 or (i.e. the natural primes congruent to ). (The natural prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
29 (number)
29 (twenty-nine) is the natural number following 28 and preceding 30. Mathematics * 29 is the tenth prime number, and the fourth primorial prime. * 29 forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. * 29 is the sum of three consecutive squares, 22 + 32 + 42. * 29 is a Lucas prime, a Pell prime, and a tetranacci number. * 29 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 29 is also the 10th supersingular prime. * None of the first 29 natural numbers have more than two different prime factors. This is the longest such consecutive sequence. * 29 is a Markov number, appearing in the solutions to ''x'' + ''y'' + ''z'' = 3''xyz'': , , , , etc. * 29 is a Perrin number, preceded in the sequence by 12, 17, 22. * 29 is the smallest positive whole number that cannot be made from the numbers , using each exactly once and using only addition, subtraction, multiplication, and ... [...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] |
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] |
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 b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Largest Known Prime
The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. A prime number is a positive integer, excluding 1, with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime. Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilise a specialised primality test that is faster than the general one. , the eight largest known primes are Mersenne primes. The last seventeen record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all 1's, since the binary form of 2''k'' − 1 is simply ''k'' 1's. Current record The record is currently held by with 24,862,048 digits, found by GIMPS in December 2018. The first and last 120 digits of its va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, June 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
101 (number)
101 (one hundred ndone) is the natural number following 100 and preceding 102. It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one". As an ordinal number, 101st (one hundred ndfirst), rather than 101th, is the correct form. In mathematics 101 is: *the 26th prime number, and the smallest above 100. *a palindromic number in base 10, and so a palindromic prime. *a Chen prime since 103 is also prime, with which it makes a twin prime pair. *a sexy prime since 107 and 113 are also prime, with which it makes a sexy prime triplet. *a unique prime, because the period length of its reciprocal is unique among primes. *an Eisenstein prime with no imaginary part and real part of the form 3n - 1. *the fifth alternating factorial. *a centered decagonal number. *the only existing prime with alternating 1s and 0s in base 10 and the largest known prime of the form 10''n'' + 1. * the number of compositions of 12 into dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
89 (number)
89 (eighty-nine) is the natural number following 88 and preceding 90. In mathematics 89 is: * the 24th prime number, following 83 and preceding 97. * a Chen prime. * a Pythagorean prime. * the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms, . * an Eisenstein prime with no imaginary part and real part of the form . * a Fibonacci number and thus a Fibonacci prime as well. The first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity ::\frac=\sum_^\infty=0.011235955\dots\ . * a Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers. ''M''89 is the 10th Mersenne prime. Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse and add process to reach a palindrome. Among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations. The palindrom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
83 (number)
83 (eighty-three) is the natural number following 82 and preceding 84. In mathematics 83 is: * the sum of three consecutive primes (23 + 29 + 31). * the sum of five consecutive primes (11 + 13 + 17 + 19 + 23). * the 23rd prime number, following 79 (of which it is also a cousin prime) and preceding 89. * a Sophie Germain prime. * a safe prime. * a Chen prime. * an Eisenstein prime with no imaginary part and real part of the form 3''n'' − 1. * a highly cototient number. * there number of primes that are right-truncatable. * a super-prime, because 23 is prime. In science Chemistry *The atomic number of bismuth (Bi) Astronomy * Messier object M83, a magnitude 8.5 spiral galaxy in the constellation Hydra, also known as the Southern Pinwheel Galaxy *The New General Catalogue object NGC 83, a magnitude 12.3 elliptical galaxy in the constellation Andromeda In religion Judaism * When someone reaches 83 they may celebrate a second bar mitzvah In music * M83 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
71 (number)
71 (seventy-one) is the natural number following 70 and preceding 72. __TOC__ In mathematics 71 is: *the 20th prime number. The next is 73, with which it composes a twin prime. *a permutable prime and emirp with 17. *is the largest number which occurs as a prime factor of an order of a sporadic simple group. *the sum of three consecutive primes: 19, 23 and 29. *a centered heptagonal number. *an Eisenstein prime with no imaginary part and real part of the form 3''n'' – 1. *a Pillai prime, since 9! + 1 is divisible by 71 but 71 is not one more than a multiple of 9. *the largest (15th) supersingular prime, which is also a Chen prime. *part of the last known pair (71, 7) of Brown numbers, since 712 = 7! + 1. *the twenty-third term of the Euclid–Mullin sequence, as it is the least prime factor of one more than the product of the first twenty-two terms. *the smallest positive integer ''d'' such that the imaginary quadratic fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
59 (number)
59 (fifty-nine) is the natural number following 58 and preceding 60. In mathematics Fifty-nine is the 17th prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary part and real part of the form . Since is divisible by 59 but 59 is not one more than a multiple of 15, 59 is a Pillai prime. It is also a highly cototient number. There are 59 stellations of the regular icosahedron, inclusive of the icosahedron. 59 is one of the factors that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13 # + 1 = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509 = 30031. 59 is the highest integer a single symbol may represent in the Sexagesimal system. As 17 is prime, 59 is a super-prime. The number 59 takes 3 iterations of the "reverse and add" process to form the palindrome 1111. All smaller integers (1 through 58) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
