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Proth Prime
A Proth number is a natural number ''N'' of the form N = k \times 2^n+1 where ''k'' and ''n'' are positive integers, ''k'' is odd and 2^n > k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. The first few Proth primes are :3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (). It is still an open question whether an infinite number of Proth primes exist. It was shown in 2022 that the reciprocal sum of Proth primes converges to a real number near 0.747392479, substantially less than the value of 1.093322456 for the reciprocal sum of Proth numbers. The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude. Definition A Proth number takes the form N=k 2^n +1 where ''k'' and ''n'' are positive integers, k is odd and 2^n>k. A Proth prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
François Proth
François Proth (22 March 1852 – 21 January 1879) was a French self-taught mathematician farmer who lived in Vaux-devant-Damloup near Verdun, France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan .... He stated four primality-related theorems. The most famous of these, Proth's theorem, can be used to test whether a Proth number (a number of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2''n'') is prime. The numbers passing this test are called Proth primes; they continue to be of importance in the computational search for large prime numbers. Proth also formulated [...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 should 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–Eule ... [...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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesel Problem
In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the form is instead k\times2^n+1, then ''k'' is a Sierpiński number. Riesel problem In 1956, Hans Riesel showed that there are an infinite number of integers ''k'' such that k\times2^n-1 is not prime for any integer ''n''. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any ''k'' less than 509203, it is conjectured to be the smallest Riesel number. To check if there are ''k'' ''k'') :2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woodall Prime
In number theory, a Woodall number (''W''''n'') is any natural number of the form :W_n = n \cdot 2^n - 1 for some natural number ''n''. The first few Woodall numbers are: :1, 7, 23, 63, 159, 383, 895, … . History Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly defined Cullen numbers. Woodall primes Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents ''n'' for which the corresponding Woodall numbers ''W''''n'' are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... ; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... . In 1976 Christopher Hooley showed that almost all Cullen numbers are composite. In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thabit Prime
In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 \cdot 2^n - 1 for a non-negative integer ''n''. The first few Thabit numbers are: : 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... The 9th century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers. Properties The binary representation of the Thabit number 3·2''n''−1 is ''n''+2 digits long, consisting of "10" followed by ''n'' 1s. The first few Thabit numbers that are prime (Thabit primes or 321 primes): :2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... , there are 68 known prime Thabit numbers. Their ''n'' values are: :0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas–Lehmer–Riesel Test
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form with odd . The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form. For numbers of the form ( Proth numbers), either application of Proth's theorem (a Las Vegas algorithm) or one of the deterministic proofs described in Brillhart–Lehmer–Selfridge 1975 (see Pocklington primality test) are used. The algorithm The algorithm is very similar to the Lucas–Lehmer test, but with a variable starting point depending on the value of . Define a sequence for all by: : u_i = u_^2-2. Then , with , is prime if and only if it divides . Finding the starting value The starting value is determined as follows. * If : if and is even, or and is odd, then divides , and there is no need to test. Otherwise, and the Lucas sequence may be used: we take u_0 = (2+\sqrt)^k+(2-\sqrt) ... [...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 should 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–Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Fermat Number
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... . If 2''k'' + 1 is prime and , then ''k'' itself must be a power of 2, so is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are , , , , and . Basic properties The Fermat numbers satisfy the following recurrence relations: : F_ = (F_-1)^+1 : F_ = F_ \cdots F_ + 2 for ''n'' ≥ 1, : F_ = F_ + 2^F_ \cdots F_ : F_ = F_^2 - 2(F_-1)^2 for . Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that and ''F''''i'' an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cullen Prime
Cullen may refer to: Places Canada *Cullen, Saskatchewan, a former hamlet in Benson No. 35 Rural Municipality Ireland * Cullen, County Cork, a village near Boherbue, County Cork * Cullen, County Tipperary, a small village in County Tipperary Scotland *Cullen, Moray, a village in Moray United States * Cullen, Kentucky *Cullen, Louisiana, a town in Webster Parish * Cullen, New York, a hamlet in the town of Warren * Cullen, Virginia * Cullen, Wisconsin, an unincorporated community in Oconto County People * Cullen (surname), an Irish surname (includes a list) * Cullen Baker (1835–1869), American criminal * Cullen Bunn (born 1971), American writer * Cullen Finnerty (1982–2013), American football player * Cullen Gillaspia (born 1995), American football player * Cullen Hightower (1923–2008), American writer * Cullen Jenkins (born 1981), American football player * Cullen Jones (born 1984), American swimmer * Cullen Landis (1896–1975), American actor and director * Culle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thabit Number
In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 \cdot 2^n - 1 for a non-negative integer ''n''. The first few Thabit numbers are: : 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... The 9th century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers. Properties The binary representation of the Thabit number 3·2''n''−1 is ''n''+2 digits long, consisting of "10" followed by ''n'' 1s. The first few Thabit numbers that are prime (Thabit primes or 321 primes): :2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... , there are 68 known prime Thabit numbers. Their ''n'' values are: :0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
