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21st
21 (twenty-one) is the natural number following 20 and preceding 22. The current century is the 21st century AD, under the Gregorian calendar. Mathematics Twenty-one is the fifth distinct semiprime, and the second of the form 3 \times q where q is a higher prime. It is a repdigit in quaternary (1114). Properties As a biprime with proper divisors 1, 3 and 7, twenty-one has a prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11, 1, 0). 21 is the first member of the second cluster of consecutive discrete semiprimes (21, 22), where the next such cluster is ( 33, 34, 35). There are 21 prime numbers with 2 digits. There are a total of 21 prime numbers between 100 and 200. 21 is the first Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. While 21 is the sixth triangular number, it is also the sum of the divisors of the first five positive integers: \begin 1 & + 2 + 3 + 4 + 5 + 6 = 21 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
35 (number)
35 (thirty-five) is the natural number following 34 (number), 34 and preceding 36 (number), 36. In mathematics 35 is the sum of the first five triangular numbers, making it a tetrahedral number. 35 is the 10th discrete semiprime (5 \times 7) and the first with 5 (number), 5 as the lowest non-unitary factor, thus being the first of the form (5.q) where q is a higher prime. 35 has two prime factors, (5 (number), 5 and 7 (number), 7) which also form its main factor pair (5 x 7) and comprise the second Twin prime, twin-prime distinct semiprime pair. The aliquot sum of 35 is 13 (number), 13, within an aliquot sequence of only one composite number (35,13 (number), 13,1 (number), 1,0) to the Prime in the 13 (number), 13-aliquot tree. 35 is the second composite number with the aliquot sum 13 (number), 13; the first being the cube 27 (number), 27. 35 is the last member of the first triple cluster of semiprimes 33 (number), 33, 34 (number), 34, 35. The second such triple distinct se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''. A decimal numeral (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in or ). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form , w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
16 (number)
16 (sixteen) is the natural number following 15 (number), 15 and preceding 17 (number), 17. It is the 4, fourth power of two. In English speech, the numbers 16 and 60 (number), 60 are sometimes confused, as they sound similar. Mathematics 16 is the ninth composite number, and a square number: 4 (number), 42 = 4 × 4 (the first non-unitary fourth-power prime number, prime of the form ''p''4). It is the smallest number with exactly five divisors, its proper divisors being , , and . Sixteen is the only integer that Equation x^y = y^x, equals ''m''''n'' and ''n''''m'', for some unequal integers ''m'' and ''n'' (m=4, n=2, or vice versa). It has this property because 2^=2\times 2. It is also equal to 32 (see tetration). The aliquot sum of 16 is 15 (number), 15, within an aliquot sequence of four composite members (16, 15 (number), 15, 9 (number), 9, 4 (number), 4, 3 (number), 3, 1 (number), 1, 0) that belong to the prime 3-aliquot tree. *Sixteen is the largest known integer , for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
12 (number)
12 (twelve) is the natural number following 11 (number), 11 and preceding 13 (number), 13. Twelve is the 3rd superior highly composite number, the 3rd colossally abundant number, the 5th highly composite number, and is divisible by the numbers from 1 (number), 1 to 4 (number), 4, and 6 (number), 6, a large number of divisors comparatively. It is central to many systems of timekeeping, including the Gregorian calendar, Western calendar and time, units of time of day, and frequently appears in the world's major religions. Name Twelve is the largest number with a monosyllable, single-syllable name in English language, English. Early Germanic languages, Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of 11 (number), eleven and twelve, the long hundred, former use of "hundred" to refer to groups of 120 (number), 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not pres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padovan Sequence
In number theory, the Padovan sequence is the integer sequence, sequence of integers ''P''(''n'') defined. by the initial values P(0) = P(1) = P(2) = 1, and the recurrence relation P(n) = P(n-2)+P(n-3). The first few values of ''P''(''n'') are :1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... The Padovan sequence is named after Richard Padovan who attributed its discovery to Netherlands, Dutch architect Hans van der Laan in his 1994 essay ''Dom. Hans van der Laan: Modern Primitive''.Richard Padovan. ''Dom Hans van der Laan: modern primitive'': Architectura & Natura Press, . The sequence was described by Ian Stewart (mathematician), Ian Stewart in his Scientific American column ''Mathematical Recreations'' in June 1996. He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics". . ''The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motzkin Number
In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. The Motzkin numbers M_n for n = 0, 1, \dots form the sequence: : 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ... Examples The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle (): : The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle (): : Properties The Motzkin numbers satisfy the recurrence relations :M_=M_+\sum_^M_iM_=\fracM_+\fracM_. The Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers: :M_n=\sum_^ \binom C_k, and inversely, :C_=\sum_^ \binom M_k This gives :\sum_^C_ = 1 + \sum_^ \binom M_. The generating function m(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagonal Number
In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o''''n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlaid so that they share one vertex (geometry), vertex. The octagonal number for ''n'' is given by the formula 3''n''2 − 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1 (number), 1, 8 (number), 8, 21 (number), 21, 40 (number), 40, 65 (number), 65, 96 (number), 96, 133 (number), 133, 176 (number), 176, 225 (number), 225, 280 (number), 280, 341, 408, 481, 560, 645, 736, 833, 936 The octagonal number for ''n'' can also be calculated by adding the square of ''n'' to twice the (''n'' − 1)th pronic number. Octagonal numbers consistently alternate parity (mathematics), parity. Octagonal numbers are occasionally referred to as "star numbers", though that term is more commonly used to refer to centered dodecagonal numbers. Appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Integers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Prime
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /math> or \Z Gaussian integers share many properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers. Gaussian integers are named after the German mathematician Carl Friedrich Gauss. Basic definitions The Gaussian integers are the set :\mathbf \, \qquad \text i^2 = -1. In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multipli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factor
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 , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
