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Cluster Prime
In number theory, a cluster prime is a prime number such that every even positive integer ''k'' ≤ p − 3 can be written as the difference between two prime numbers not exceeding . For example, the number 23 is a cluster prime because 23 − 3 = 20, and every even integer from 2 to 20, inclusive, is the difference of at least one pair of prime numbers not exceeding 23: * 5 − 3 = 2 * 7 − 3 = 4 * 11 − 5 = 6 * 11 − 3 = 8 * 13 − 3 = 10 * 17 − 5 = 12 * 17 − 3 = 14 * 19 − 3 = 16 * 23 − 5 = 18 * 23 − 3 = 20 On the other hand, 149 is not a cluster prime because 140 < 146, and there is no way to write 140 as the difference of two primes that are less than or equal to 149. By convention, 2 is not considered to be a cluster prime. The first 23 odd primes (up to 89) are all cluster primes. The first few odd primes that are not cluster primes are : 97, [...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] |
149 (number)
149 (one hundred [and] forty-nine) is the natural number between 148 (number), 148 and 150 (number), 150. In mathematics 149 is a prime number, the first prime whose difference from the previous prime is exactly 10, an emirp, and an irregular prime. After 1 and 127, it is the third smallest de Polignac number, an odd number that cannot be represented as a prime plus a power of two. More strongly, after 1, it is the second smallest number that is not a sum of two prime powers. It is a tribonacci number, being the sum of the three preceding terms, 24, 44, 81. There are exactly 149 integer points in a closed circular disk of radius 7, and exactly 149 ways of placing six queens (the maximum possible) on a 5 × 5 chess board so that each queen attacks exactly one other. The barycentric subdivision of a tetrahedron produces an abstract simplicial complex with exactly 149 simplices. See also * The year AD 149 or 149 BC * List of highways numbered 149 * References Externa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
97 (number)
97 (ninety-seven) is the natural number following 96 and preceding 98. It is a prime number and the only prime in the nineties. In mathematics 97 is: * the 25th prime number (the largest two-digit prime number in base 10), following 89 and preceding 101. * a Proth prime and a Pierpont prime as it is 3 × 25 + 1. * the eleventh member of the Mian–Chowla sequence. * a self number in base 10, since there is no integer that added to its own digits, adds up to 97. * the smallest odd prime that is not a cluster prime. * the highest two-digit number where the sum of its digits is a square. * the number of primes <= 29. * The numbers 97, 907, 9007, 90007 and 900007 are all primes, and they are all s. However, 9000007 (read as ''nine million seven'') is |
127 (number)
127 (one hundred ndtwenty-seven) is the natural number following 126 and preceding 128. It is also a prime number. In mathematics *As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known mersenne prime exponent for a Mersenne number, 2^-1, which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years. **2^-1 is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime. ** Furthermore, 127 is equal to 2^-1, and 7 is equal to 2^-1, and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime. *There are a total of 127 prime numbers between 2,000 and 3,000. *127 is also a cuban prime of the form p=\frac, x=y+1. The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
191 (number)
191 (one hundred ndninety-one) is the natural number following 190 and preceding 192. In mathematics 191 is a prime number, part of a prime quadruplet of four primes: 191, 193, 197, and 199. Because doubling and adding one produces another prime number (383), 191 is a Sophie Germain prime. It is the smallest prime that is not a full reptend prime in ''any'' base from 2 to 10; in fact, the smallest base for which 191 is a full period prime is base 19 There are many different numeral systems, that is, writing systems for expressing numbers. By culture / time period By type of notation Numeral systems are classified here as to whether they use positional notation (also known as place-value ....Wolfram MathWorldPrimitive Root/ref> See also * 191 (other) References Integers {{num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
211 (number)
211 (two hundred ndeleven) is the natural number following 210 and preceding 212. It is also a prime number. In mathematics 211 is an odd number. 211 is a primorial prime, sum of three consecutive primes (67 + 71 + 73), Chen prime, centered decagonal prime, and self prime. 211 is the smallest prime separated by 10 or more from the nearest primes (199 and 223). It is thus a balanced prime and an ''isolated prime''. 211 is a repdigit in base 14 (111). Multiplying its digits, it is still a prime (2), and adding its digits, it is square (4). Rearranging its digits, 211 becomes 121, which also is a square. Adding any two of its digits will be prime (2 or 3). 211 is a super-prime. In science and technology 2-1-1 is special abbreviated telephone number reserved in Canada and the United States as an easy-to-remember three-digit telephone number. It is meant to provide quick information and referrals to health and human service organizations for both services from charities and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
223 (number)
223 (two hundred ndtwenty-three) is the natural number following 222 and preceding 224. In mathematics 223 is a prime number. Among the 720 permutations of the numbers from 1 to 6, exactly 223 of them have the property that at least one of the numbers is fixed in place by the permutation and the numbers less than it and greater than it are separately permuted among themselves. In connection with Waring's problem, 223 requires the maximum number of terms (37 terms) when expressed as a sum of positive fifth powers, and is the only number that requires that many terms. In other fields * .223 (other), the caliber of several firearm cartridges * The years 223 and 223 BC __NOTOC__ Year 223 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Flaminus and Philus (or, less frequently, year 531 '' Ab urbe condita''). The denomination 223 BC for this year has bee ... * The number of synodic months of a Saros Referen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
227 (number)
227 (two hundred ndtwenty-seven) is the natural number between 226 and 228. It is also a prime number. In mathematics 227 is a twin prime and the start of a prime triplet (with 229 and 233). It is a safe prime, as dividing it by two and rounding down produces the Sophie Germain prime 113. It is also a regular prime, a Pillai prime, a Stern prime, and a Ramanujan prime. 227 and 229 form the first twin prime pair for which neither is a cluster prime. The 227th harmonic number is the first to exceed six. There are 227 different connected graphs with eight edges, and 227 independent sets in a 3 × 4 grid graph In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a latti .... References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
229 (number)
229 (two hundred ndtwenty-nine) is the natural number following 228 and preceding 230. In mathematics It is a prime number, and a regular prime. It is also a full reptend prime, meaning that the decimal expansion of the unit fraction 1/229 repeats periodically with as long a period as possible. With 227 it is the larger of a pair of twin primes, and it is also the start of a sequence of three consecutive squarefree numbers. It is the smallest prime that, when added to the reverse of its decimal representation, yields another prime: 229 + 922 = 1151. There are 229 cyclic permutations of the numbers from 1 to 7 in which none of the numbers is mapped to its successor (mod 7), 229 rooted tree structures formed from nine carbon atoms, and 229 triangulations of a polygon formed by adding three vertices to each side of a triangle. There are also 229 different projective configurations of type (123123), in which twelve points and twelve lines meet with three lines ... [...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] |
Twin Prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Set (combinatorics)
In combinatorial mathematics, a large set of positive integers :S = \ is one such that the infinite sum of the reciprocals :\frac+\frac+\frac+\frac+\cdots diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges. Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions. Examples * Every finite subset of the positive integers is small. * The set \ of all positive integers is known to be a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form ''an'' + ''b'' with ''a'' ≥ 1, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) is a large set. * The set of square numbers is small (see Basel problem). So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
