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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 is 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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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 its digits gives another prime number (701) In other fields 107 is also: * The emergency telephone number in Argentina and Cape Town. * The telephone of the police in Hungary. In sports * The 107% rule, a Formula One Sporting Regulation in operation from 1996 to 2002 and 2011 onward. * The number 107 is also associated with the Timbers Army supporters group of the Portland Timbers socc ... [...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 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. Th ... [...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 In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ... of a 10 × 10 (0,1)-matrix. In other fields * Calf 269 was a calf that rose to fame after being rescued by Israeli activists in 2012. As a result, numerous people branded the number "269" into their bodies over 2012 and 2013. References See also * 269 AD * 269 BC * 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 Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ... has exactly 251 square submatrices. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
239 (number)
239 (two hundred ndthirty-nine) is the natural number following 238 and preceding 240. Properties 239 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 a factor of the repdigit 1111111, with the other prime factor being 4649. 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 entry 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 max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
227 (number)
227 (two hundred ndtwenty-seven) is the natural number following 226 and preceding 228. It is also a prime number. In mathematics 227 is a 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 or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ..., and the start of a prime triplet. 227 is a safe prime, a regular prime, a Pillai prime, a Stern prime, and a Ramanujan prime. 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, *a lucky prime, *a left- truncatable prime, and a left-and-right-truncatable prime. 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. See also * 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 Flaminius and Philus (or, less frequently, year 531 ''Ab urbe condita''). The denomination 223 BC for this year has bee ... References Integers {{Num ... [...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 seven 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 In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ... by a non-crossing set of its dia ... [...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 repetend 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 "A ''base'' is a natural number B whose ''powers'' (B multiplied by itself some number of times) are specially designated wit ....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 In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs .... 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. See also * References External links Integers {{Num-stub ... [...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 the 38th prime number and 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. As approximations, \pi \approx \approx 3.1411..., and e \approx \approx 2.7166\dots 163 is a permutable prime in base 12, which it is written as 117, the permutations of its digits are 171 and 711, the two numbers in base 12 are 229 and 1021 in base 10, both of which are prime. The function f(n) = n^2 - n + 41 gives prime values for all values of n between 0 and 39, while for n < 3000 approximately half of all values are prime. 163 ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
