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6 (number)
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. In mathematics Six is the smallest positive integer which is neither a square number nor a prime number; it is the second smallest composite number, behind 4; its proper divisors are , and . Since 6 equals the sum of its proper divisors, it is a perfect number; 6 is the smallest of the perfect numbers. It is also the smallest Granville number, or \mathcal-perfect number. As a perfect number: *6 is related to the Mersenne prime 3, since . (The next perfect number is 28.) *6 is the only even perfect number that is not the sum of successive odd cubes. *6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, . Six is the only number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". Six is a ...
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Senary
A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like decimal, it is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to senary. In turn, the senary logic refers to an extension of Jan Łukasiewicz's and Stephen Cole Kleene's ternary logic systems adjusted to explain the logic of statistical tests and missing data patterns in sciences using empirical methods. Formal definition The standard set of digits in senary is given by \mathcal_6 = \lbrace 0, 1, 2, 3, 4, 5\rbrace, with a linear order 0 < 1 < 2 < 3 < 4 < 5. Let \mathcal_6^* be the Kleene closure of \mathcal_6
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Khmer Numerals
Khmer numerals are the numerals used in the Khmer language. They have been in use since at least the early 7th century, with the earliest known use being on a stele dated to AD 604 found in Prasat Bayang, near Angkor Borei, Cambodia. Numerals Having been derived from the Hindu–Arabic numeral system, Hindu numerals, modern Khmer numerals also represent a decimal positional notation system. It is the script with the first extant material evidence of Zero#Zero as a decimal digit, zero as a numerical figure, dating its use back to the seventh century, two centuries before its certain use in India. Old Khmer, or Angkorian Khmer, also possessed separate symbols for the numbers 10, 20, and 100. Each multiple of 20 or 100 would require an additional stroke over the character, so the number 47 was constructed using the 20 symbol with an additional upper stroke, followed by the symbol for number 7. This inconsistency with its decimal, decimal system suggests that spoken #Angkori ...
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Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem as ...
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Granville Number
In mathematics, specifically number theory, Granville numbers, also known as \mathcal-perfect numbers, are an extension of the perfect numbers. The Granville set In 1996, Andrew Granville proposed the following construction of a set \mathcal: :Let 1\in\mathcal, and for any integer n larger than 1, let n\in if ::\sum_ d \leq n. A Granville number is an element of \mathcal for which equality holds, that is, n is a Granville number if it is equal to the sum of its proper divisors that are also in \mathcal. Granville numbers are also called \mathcal-perfect numbers. General properties The elements of \mathcal can be -deficient, -perfect, or -abundant. In particular, 2-perfect numbers are a proper subset of \mathcal. S-deficient numbers Numbers that fulfill the strict form of the inequality in the above definition are known as \mathcal-deficient numbers. That is, the \mathcal-deficient numbers are the natural numbers for which the sum of their divisors in \mathcal is strictly ...
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Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they ar ...
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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 alway ...
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Square Number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, \sqrt = 3, so 9 is a square ...
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Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p-1 for positive integer p—what is now called a Mersenne prime. Two millen ...
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Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 ×  7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, ...
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Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as '' nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by succ ...
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Malayalam Numerals
Malayalam numerals are the numeral system of the Malayalam script used by Malayalam in Kerala. It is one of several Indian numeral systems. This system is archaic and nowadays the Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ... is used commonly. However it is still found in the Malayalam Bible denoting the chapters. Base numbers Below is a list of Malayalam numerals with their Hindu–Arabic equivalents as well as their respective Malayalam translations and transliterations. Originally, a number like "11" would have been written as "൰൧" and not "൧൧" to match the Malayalam word for 11 and "10,00,000" as "൰൱൲" similar to the Tamil numeral system. Later on this system got reformed to be more similar to the Hindu–Arabic numerals so ...
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Saraiki Language
Saraiki ( '; also spelt Siraiki, or Seraiki) is an Indo-Aryan language of the Lahnda group, spoken by 26 million people primarily in the south-western half of the province of Punjab in Pakistan. It was previously known as Multani, after its main dialect. Saraiki has partial mutual intelligibility with Standard Punjabi, and it shares with it a large portion of its vocabulary and morphology. At the same time in its phonology it is radically different (particularly in the lack of tones, the preservation of the voiced aspirates and the development of implosive consonants), and has important grammatical features in common with the Sindhi language spoken to the south. The Saraiki language identity arose in the 1960s, encompassing more narrow local earlier identities (like Multani, Derawi or Riasati), and distinguishing itself from broader ones like that of Punjabi. Name The present extent of the meaning of ' is a recent development, and the term most probably gained its cur ...
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