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Strong Prime
In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory. Definition in number theory In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (''p'', ''p'', ''p'', ...) = (2, 3, 5, ...), ''p'' is a strong prime if . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime. The first few strong primes are : 11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 . In a twin prime pair (''p'', ''p'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
107 (number)
107 (one hundred ndseven) is the natural number following 106 and preceding 108. In mathematics 107 is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime. Plugged into the expression 2^p - 1, 107 yields 162259276829213363391578010288127, a Mersenne prime. 107 is itself a safe prime. It is the fourth Busy beaver number, the maximum number of steps that any Turing machine with 2 symbols and 4 states can make before eventually halting. It is the number of triangle-free graphs on 7 vertices. It is the ninth emirp, because reversing it's digits gives another prime number (701) In other fields As "one hundred ''and'' seven", it is the smallest positive integer requiring six syllables in English (without the "and" it only has five syllables and seventy-seven is a smaller 5-syllable number). 107 is also: * The atomic number of bohrium. * The emergency telephone number in Argentina and Cape Town. * The telephone of the poli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
277 (number)
277 (two hundred ndseventy-seven) is the natural number following 276 and preceding 278. Mathematical properties 277 is the 59th prime number, and is a regular prime. It is the smallest prime ''p'' such that the sum of the inverses of the primes up to ''p'' is greater than two. Since 59 is itself prime, 277 is a super-prime. 59 is also a super-prime (it is the 17th prime), as is 17 (the 7th prime). However, 7 is the fourth prime number, and 4 is not prime. Thus, 277 is a super-super-super-prime but not a super-super-super-super-prime. It is the largest prime factor of the Euclid number 510511 = 2 × 3 × 5 × 7 × 11 × 13 × 17 + 1. As a member of the lazy caterer's sequence, 277 counts the maximum number of pieces obtained by slicing a pancake with 23 straight cuts. 277 is also a Perrin number, and as such counts the number of maximal independent sets in an icosagon. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
269 (number)
269 (two hundred ndsixty-nine) is the natural number between 268 and 270. It is also a prime number. In mathematics 269 is a twin prime, and a Ramanujan prime. It is the largest prime factor of 9! + 1 = 362881, and the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix. In media * "Hawkmoon 269", pop song by U2 See also * Area code 269 * Calf 269 Calf 269 is a bull who was rescued as a calf by anonymous activists, days before his planned slaughter. He was born at an Israeli facility in the vicinity of Azor, a town on the outskirts of Tel Aviv. The slaughter was scheduled for June 2013. H ..., animal liberation movement * List of highways numbered 269 References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
251 (number)
251 (two hundred ndfifty-one) is the natural number between 250 and 252. It is also a prime number. In mathematics 251 is: *a Sophie Germain prime. *the sum of three consecutive primes (79 + 83 + 89) and seven consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47). *a Chen prime. *an Eisenstein prime with no imaginary part. *a de Polignac number, meaning that it is odd and cannot be formed by adding a power of two to a prime number. *the smallest number that can be formed in more than one way by summing three positive cubes:251 = 2^3 + 3^3 + 6^3 = 1^3 + 5^3 + 5^3. Every 5 × 5 matrix has exactly 251 square submatrices. In science *The average atomic mass and most stable isotope of Californium, which has a half life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable at ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
239 (number)
239 (two hundred ndthirty-nine) is the natural number following 238 and preceding 240. In mathematics It is a prime number. The next is 241, with which it forms a pair of twin primes; hence, it is also a Chen prime. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime. It is an Eisenstein prime with no imaginary part and real part of the form 3''n'' − 1 (with no exponentiation implied). 239 is also a happy number. 239 is the smallest positive integer ''d'' such that the imaginary quadratic field Q() has class number = 15. HAKMEM (incidentally AI memo 239 of the MIT AI Lab) included an item on the properties of 239, including these: * When expressing 239 as a sum of square numbers, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positive cubes (23 is the only other such integer), and the maximum number (19) of fourth powers. * 239/ 169 is a convergent of the continued fraction of ... [...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] |
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] |
197 (number)
197 (one hundred ndninety-seven) is the natural number following 196 and preceding 198. In mathematics * 197 is a prime number, the third of a prime quadruplet: 191, 193, 197, 199 * 197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41, and is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 * 197 is a centered heptagonal number, a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers * 197 is a Schröder–Hipparchus number, counting for instance the number of ways of subdividing a heptagon by a non-crossing set of its diagonals. In other fields 197 is also: * A police emergency telephone number in Tunisia * Number enquiry telephone number in Nepal * a song by Norwegian alternative rock group Major Parkinson Major Parkinson is a Norwegian rock group currently based in Berge ... [...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] |
179 (number)
179 (one hundred ndseventy-nine) is the natural number following 178 and preceding 180. In mathematics 179 is part of the Cunningham chain of prime numbers 89, 179, 359, 719, 1439, 2879, in which each successive number is two times the previous number, plus one. Among Cunningham chains of this length, this one has the smallest numbers. Because 179 is neither the start nor the end of this chain, it is both a safe prime and a Sophie Germain prime. It is also a super-prime number, because it is the 41st smallest prime and 41 is also prime. Since 971 (the digits of 179 reversed) is prime, 179 is an emirp. In other fields Astronomers have suggested that sunspot frequency undergoes a cycle of approximately 179 years in length. See also * AD 179 and 179 BC __NOTOC__ Year 179 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Flaccus and Fulvianus (or, less frequently, year 575 ''Ab urbe condita''). The denomination 179 BC for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
163 (number)
163 (one hundred ndsixty-three) is the natural number following 162 and preceding 164. In mathematics 163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes. 163 is a lucky prime and a fortunate number. 163 is a strictly non-palindromic number, since it is not palindromic in any base between base 2 and base 161. Given 163, the Mertens function returns 0, it is the fourth prime with this property, the first three such primes are 2, 101 and 149. 163 figures in an approximation of π, in which \pi \approx \approx 3.1411. 163 figures in an approximation of ''e'', in which e \approx \approx 2.7166\dots. 163 is a Heegner number, the largest of the nine such numbers. That is, the ring of integers of the field \mathbb(\sqrt) has unique factorization for a=163. The only other such integers are a = 1, 2, 3, 7, 11, 19, 43, 67. 163 is the number of -independent McKay-Thompson series for the monster group. This fact about 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
