Super-prime

	Super-prime Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched with ordinal numbers, starting with prime number 2 matched with ordinal number 1, then the primes matched with prime ordinal numbers are the super-primes. The subsequence begins :3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... . That is, if ''p''(''n'') denotes the ''n''th prime number, the numbers in this sequence are those of the form ''p''(''p''(''n'')). In 1975, Robert Dressler and Thomas Parker used a computer-aided proof (based on calculations involving the subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a partial order. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as \langle B,C,D \rangle, from \langle A,B,C,D,E,F \rangle, is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "''a''", "''ap''", "''al''", "''ae''", "''app''", "''apl''", "''ape''", "''ale''", "''appl''", "''appe''", "''aple''", "''apple''", "''p''", "''pp''", "''pl''", "''pe''", "''ppl''", "''ppe''", " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Prime Number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 factorization, 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 primality test, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Subset Sum Problem The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Bertrand's Postulate In number theory, Bertrand's postulate is the theorem that for any integer n > 3, there exists at least one prime number p with :n < p < 2n - 2. A less restrictive formulation is: for every $n > 1$ , there is always at least one prime $p$ such that : $n < p < 2n.$ Another formulation, where $p_n$ is the $n$ -th prime, is: for $n \ge 1$ : $p_ < 2p_n.$ This statement was first conjectured in 1845 by Joseph Bertrand (1822–1900). Bertrand himself verified his statement for all integers $2 \le n \le 3\,000\,000$ . His conjecture was completely Proof of Bertrand's postulate, proved by Pafnuty Chebyshev, Chebyshev (1821–1894) in 1852 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with $\pi(x)$ , the prime-counting function (number of primes less than or equal to $x$ ): ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Small Set (combinatorics) In combinatorial mathematics, a large set of positive integers :S = \ is one such that the infinite sum of the reciprocals :\frac+\frac+\frac+\frac+\cdots diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges. Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions. Examples * Every finite subset of the positive integers is small. * The set \ of all positive integers is a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form ''an'' + ''b'' with ''a'' ≥ 1, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) is a large set. * The set of square numbers is small (see Basel problem). So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive integer va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Palindromic Prime In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are: : 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. For any base, almost all palindromic numbers are composite, i.e. the ratio between palindromic composites and all palindromes less than ''n'' tends to 1. A few decorative examples do however exist; in base 10 the following are primes: 11, 122333221, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]