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11th Century BC In Greece
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables. Name "Eleven" derives from the Old English ', which is first attested in Bede's late 9th-century ''Ecclesiastical History of the English People''. It has cognates in every Germanic language (for example, German ), whose Proto-Germanic ancestor has been reconstructed as , from the prefix (adjectival "one") and suffix , of uncertain meaning. It is sometimes compared with the Lithuanian ', though ' is used as the suffix for all numbers from 11 to 19 (analogously to "-teen"). The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as . This was formerly thought to be derived from Proto-Germanic ("ten"); it is now sometimes connected with or ("left; remaining"), with the implicit meaning that "one is left" after counting to ten.''Oxford English Dictio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which alw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proto-Germanic Language
Proto-Germanic (abbreviated PGmc; also called Common Germanic) is the reconstructed proto-language of the Germanic branch of the Indo-European languages. Proto-Germanic eventually developed from pre-Proto-Germanic into three Germanic branches during the fifth century BC to fifth century AD: West Germanic, East Germanic and North Germanic, which however remained in contact over a considerable time, especially the Ingvaeonic languages (including English), which arose from West Germanic dialects and remained in continued contact with North Germanic. A defining feature of Proto-Germanic is the completion of the process described by Grimm's law, a set of sound changes that occurred between its status as a dialect of Proto-Indo-European and its gradual divergence into a separate language. As it is probable that the development of this sound shift spanned a considerable time (several centuries), Proto-Germanic cannot adequately be reconstructed as a simple node in a tree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Prime
A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality :p_n^2 > p_ \cdot p_ for all 1 ≤ ''i'' ≤ ''n''−1, where ''pk'' is the ''k''th prime. Example: the first primes are 2, 3, 5, 7 and 11. Since for 5 both the conditions :5^2 >3 \cdot 7 :5^2 >2 \cdot 11 are fulfilled, 5 is a good prime. There are infinitely many good primes. The first good primes are: : 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967 . An alternative version takes only ''i'' = 1 in the definition. With that there are more good primes: : 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 227, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Prime
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737. Like all rational numbers, the reciprocals of primes have repeating decimal representations. In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes. Contemporaneously, William Shanks (1812–1882) calculated numerous reciprocals of primes and their repeating periods, and published two papers "On Periods in the Reciprocals of Primes" in 1873 and 1874. In 1874 he also published a table of primes, and the periods of their reciprocals, up to 20,000 (with help from and "communicated by the Rev. George Salmon"), and pointed out the errors in previous tables by three other authors. Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. For a prime , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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''&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repunit Prime
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreations in the Theory of Numbers''. A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of March 2022, the largest known prime number , the largest probable prime ''R''8177207 and the largest elliptic curve primality prime ''R''49081 are all repunits. Definition The base-''b'' repunits are defined as (this ''b'' can be either positive or negative) :R_n^\equiv 1 + b + b^2 + \cdots + b^ = \qquad\mbox, b, \ge2, n\ge1. Thus, the number ''R''''n''(''b'') consists of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n'' = 1 and ''n'' = 2 are :R_1^ 1 \qquad \text \qquad R_2^ b+1\qquad\text\ , b, \ge2. ... [...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] |
13 (number)
13 (thirteen) is the natural number following 12 and preceding 14. Strikingly folkloric aspects of the number 13 have been noted in various cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition. In mathematics The number 13 is the sixth prime number. It is a twin prime with 11, as well as a cousin prime with 17. It is the second Wilson prime, of three known (the others being 5 and 563), and the smallest emirp in decimal. 13 is: *The second star number: *The third centered square number: * A happy number and a lucky number. *A Fibonacci number, preceded by 5 and 8. *The smallest number whose fourth power can be written as a sum of two consecutive square numbers (1192 + 120 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palindromic Number
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palindromic'' is derived from palindrome, which refers to a word (such as ''rotor'' or ''racecar'') whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers (in decimal) are: : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, … . Palindromic numbers receive most attention in the realm of recreational mathematics. A typical problem asks for numbers that possess a certain property ''and'' are palindromic. For instance: * The palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, ... . * The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... . It is obvious that in any radix, base there are Infinite set, infin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Old Norse
Old Norse, Old Nordic, or Old Scandinavian, is a stage of development of North Germanic languages, North Germanic dialects before their final divergence into separate Nordic languages. Old Norse was spoken by inhabitants of Scandinavia and their Viking expansion, overseas settlements and chronologically coincides with the Viking Age, the Christianization of Scandinavia and the consolidation of Scandinavian kingdoms from about the 7th to the 15th centuries. The Proto-Norse language developed into Old Norse by the 8th century, and Old Norse began to develop into the modern North Germanic languages in the mid-to-late 14th century, ending the language phase known as Old Norse. These dates, however, are not absolute, since written Old Norse is found well into the 15th century. Old Norse was divided into three dialects: Old West Norse, ''Old West Norse'' or ''Old West Nordic'' (often referred to as ''Old Norse''), Old East Norse, ''Old East Norse'' or ''Old East Nordic'', and ''Ol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Old Saxon
Old Saxon, also known as Old Low German, was a Germanic language and the earliest recorded form of Low German (spoken nowadays in Northern Germany, the northeastern Netherlands, southern Denmark, the Americas and parts of Eastern Europe). It is a West Germanic language, closely related to the Anglo-Frisian languages. It is documented from the 8th century until the 12th century, when it gradually evolved into Middle Low German. It was spoken throughout modern northwestern Germany, primarily in the coastal regions and in the eastern Netherlands by Saxons, a Germanic tribe that inhabited the region of Saxony. It partially shares Anglo-Frisian's (Old Frisian, Old English) Ingvaeonic nasal spirant law which sets it apart from Low Franconian and Irminonic languages, such as Dutch, Luxembourgish and German. The grammar of Old Saxon was fully inflected with five grammatical cases ( nominative, accusative, genitive, dative, and instrumental), three grammatical numbers ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Old Frisian
Old Frisian was a West Germanic language spoken between the 8th and 16th centuries along the North Sea coast, roughly between the mouths of the Rhine and Weser rivers. The Frisian settlers on the coast of South Jutland (today's Northern Friesland) also spoke Old Frisian, but there are no known medieval texts from this area. The language of the earlier inhabitants of the region between the Zuiderzee and Ems River (the Frisians mentioned by Tacitus) is attested in only a few personal names and place-names. Old Frisian evolved into Middle Frisian, spoken from the 16th to the 19th century. In the early Middle Ages, Frisia stretched from the area around Bruges, in what is now Belgium, to the Weser River in northern Germany . At the time, the Frisian language was spoken along the entire southern North Sea coast. This region is referred to as Greater Frisia or Magna Frisia, and many of the areas within it still treasure their Frisian heritage. However, by 1300, their territory ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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