|
Composite Numbers
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, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number Cuisenaire Rods 10
Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ** Dental composite, a substance used to fill cavities in teeth ** Composite armor, a type of tank armor * Alloy, a mixture of a metal and another element * Mixture, the combination of several different substances without chemical reaction Mathematics * Composite number, a positive integer that has at least one factor other than one or itself Science * Composite particle, a particle which is made up of smaller particles * ''Compositae'' or "composite family" of flowering plants * Composite volcano, a layered conical volcano * Compositing, another name for superposed epoch analysis, a statistical method used to analyze time series involving multiple events Technology * Compositing, combining of visual elements from separate sources into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powerful Number
A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a square and a cube, that is, a number ''m'' of the form ''m'' = ''a''2''b''3, where ''a'' and ''b'' are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers ''powerful''. The following is a list of all powerful numbers between 1 and 1000: :1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... . Equivalence of the two definitions If ''m'' = ''a''2''b''3, then every prime in the prime factorization of ''a'' appears in the prime factorization of ''m'' with an exponent of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addison-Wesley
Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles through the O'Reilly Online Learning e-reference service. Addison-Wesley's majority of sales derive from the United States (55%) and Europe (22%). The Addison-Wesley Professional Imprint produces content including books, eBooks, and video for the professional IT worker including developers, programmers, managers, system administrators. Classic titles include ''The Art of Computer Programming'', ''The C++ Programming Language'', ''The Mythical Man-Month'', and ''Design Patterns''. History Lew Addison Cummings and Melbourne Wesley Cummings founded Addison-Wesley in 1942, with the first book published by Addison-Wesley being Massachusetts Institute of Technology professor Francis Weston Sears' ''Mechanics''. Its first computer book was ''Progra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Prime Factors
The tables contain the prime factorization of the natural numbers from 1 to 1000. When ''n'' is a prime number, the prime factorization is just ''n'' itself, written in bold below. The number 1 is called a unit. It has no prime factors and is neither prime nor composite. Properties Many properties of a natural number ''n'' can be seen or directly computed from the prime factorization of ''n''. *The multiplicity of a prime factor ''p'' of ''n'' is the largest exponent ''m'' for which ''pm'' divides ''n''. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since ''p'' = ''p''1). The multiplicity of a prime which does not divide ''n'' may be called 0 or may be considered undefined. *Ω(''n''), the big Omega function, is the number of prime factors of ''n'' counted with multiplicity (so it is the sum of all prime factor multiplicities). *A prime number has Ω(''n'') = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sieve Of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime.Horsley, Rev. Samuel, F. R. S., "' or, The Sieve of Eratosthenes. Being an account of his method of finding all the Prime Numbers,''Philosophical Transactions'' (1683–1775), Vol. 62. (1772), pp. 327–347 This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes. The earliest known reference to the sieve ( grc, κόσκινον Ἐρατοσθένους, ''kóskinon Erat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Representation Of A Positive Integer
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example, : 1200 = 2^4 \cdot 3^1 \cdot 5^2 = (2 \cdot 2 \cdot 2 \cdot 2) \cdot 3 \cdot (5 \cdot 5) = 5 \cdot 2 \cdot 5 \cdot 2 \cdot 3 \cdot 2 \cdot 2 = \ldots The theorem says two things about this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example, 12 = 2 \cdot 6 = 3 \cdot 4). This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rough Number
A ''k''-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to ''k''. ''k''-roughness has alternately been defined as requiring all prime factors to strictly exceed ''k''.p. 130, Naccache and Shparlinski 2009. Examples (after Finch) #Every odd positive integer is 3-rough. #Every positive integer that is congruent to 1 or 5 mod 6 is 5-rough. #Every positive integer is 2-rough, since all its prime factors, being prime numbers, exceed 1. See also * Buchstab function, used to count rough numbers * Smooth number Notes References * Finch's definition from Number Theory Archives* "Divisibility, Smoothness and Cryptographic Applications", D. Naccache and I. E. Shparlinski, pp. 115–173 in ''Algebraic Aspects of Digital Communications'', eds. Tanush Shaska and Engjell Hasimaj, IOS Press, 2009, . The On-Line Encyclopedia of Integer Sequences (OEIS) lists ''p''-rough numbers for small ''p'': * 2-rough numbers: A00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers. Definition A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, resp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pronic Number
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers. The first few pronic numbers are: : 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 … . Letting P_n denote the pronic number n(n+1), we have P_ = P_. Therefore, in discussing pronic numbers, we may assume that n\geq 0 without loss of generality, a convention that is adopted in the following sections. As figurate numbers The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's ''Metaphysics'', and their discovery has been attributed much earlier to the Pythagoreans.. As a kind of figurate number, the pronic numbers are somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Number
__FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. The late mathematician Jean-Pierre Kahane has suggested that Plato must have known about highly composite numbers as he deliberately chose 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. Ramanujan wrote and titled his paper on the subject in 1915. Examples The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate superior highly composite numbers. The divisors of the first 15 highly composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
42 (number)
42 (forty-two) is the natural number that follows 41 (number), 41 and precedes 43 (number), 43. Mathematics Forty-two (42) is a pronic number and an abundant number; its prime factorization (2\times 3\times 7) makes it the second sphenic number and also the second of the form (2\times 3\times r). Additional properties of the number 42 include: * It is the number of isomorphism classes of all simple and oriented directed graphs on 4 vertices. In other words, it is the number of all possible outcomes (up to isomorphism) of a tournament consisting of 4 teams where the game between any pair of teams results in three possible outcomes: the first team wins, the second team wins, or there is a draw. The group stage of the FIFA World cup is a good example. * It is the third primary pseudoperfect number. * It is a Catalan number. Consequently, 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered bina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
