Concepts inSums of divisors, perfect numbers, and factoring
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities.
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Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. ¿1(n) = 2n.
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Divisor
In mathematics, a divisor of an integer, also called a factor of, is an integer which divides without leaving a remainder.
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RP (complexity)
In complexity theory, RP ("randomized polynomial time") is the complexity class of problems for which a probabilistic Turing machine exists with these properties: It always runs in polynomial time in the input size If the correct answer is NO, it always returns NO If the correct answer is YES, then it returns YES with probability at least 1/2 (otherwise, it returns NO). In other words, the algorithm is allowed to flip a truly random coin while it is running.
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Bounded-error probabilistic polynomial
In computational complexity theory, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1/3 for all instances.
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Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer. When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort concluded in 2009 by several researchers factored a 232-digit number, utilizing hundreds of machines over a span of 2 years.
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Complexity class
In computational complexity theory, a complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form: the set of problems that can be solved by an abstract machine M using O(f) of resource R, where n is the size of the input.
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Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. A later notion is that of a nominal number, which is used only for naming. Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory.
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