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Sum-free Sequence
In mathematics, a sum-free sequence is an increasing sequence of positive integers, :a_1, a_2, a_3, \ldots, such that no term a_n can be represented as a sum of any subset of the preceding elements of the sequence. This differs from a sum-free set, where only pairs of sums must be avoided, but where those sums may come from the whole set rather than just the preceding terms. Example The powers of two, :1, 2, 4, 8, 16, ... form a sum-free sequence: each term in the sequence is one more than the sum of all preceding terms, and so cannot be represented as a sum of preceding terms. Sums of reciprocals A set of integers is said to be small if the sum of its reciprocals converges to a finite value. For instance, by the prime number theorem, the prime numbers are not small. proved that every sum-free sequence is small, and asked how large the sum of reciprocals could be. For instance, the sum of the reciprocals of the powers of two (a geometric series) is two. If R denotes the ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum-free Set
In additive combinatorics and number theory, a subset ''A'' of an abelian group ''G'' is said to be sum-free if the sumset ''A'' + ''A'' is disjoint from ''A''. In other words, ''A'' is sum-free if the equation a + b = c has no solution with a,b,c \in A. For example, the set of odd numbers is a sum-free subset of the integers, and the set forms a large sum-free subset of the set . Fermat's Last Theorem is the statement that, for a given integer ''n'' > 2, the set of all nonzero ''n''th powers of the integers is a sum-free set. Some basic questions that have been asked about sum-free sets are: * How many sum-free subsets of are there, for an integer ''N''? Ben Green has shown that the answer is O(2^), as predicted by the Cameron–Erdős conjecture. * How many sum-free sets does an abelian group ''G'' contain?Ben Green and Imre RuzsaSum-free sets in abelian groups 2005. * What is the size of the largest sum-free set that an abelian group ''G'' contains? A sum-free set is sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powers Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hierarchy, is exactly equal to H_(1). Powers of two with non-negative exponents are integers: , , and is two multiplied by itself times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... By comparison, powers of two with negative exponents are fractions: for positive integer , is one half multiplied by itself times. Thus the first few negative powers of 2 are , , , , etc. Sometimes these are called ''inverse powers of two'' because each is the multiplicative inverse of a positive power of two. Base of the binary numeral system Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always h ... [...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] |
Multiplicative Inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rational number, fraction ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the Function (mathematics), function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an Involution (mathematics), involution). Multiplying by a number is the same as Division (mathematics), dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent Series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of powers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
