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Prime K-tuple
In number theory, a prime -tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a -tuple , the positions where the -tuple matches a pattern in the prime numbers are given by the set of integers for which all of the values are prime. Typically the first value in the -tuple is 0 and the rest are distinct positive even numbers. Named patterns Several of the shortest ''k''-tuples are known by other common names: OEIS sequence A257124 covers 7-tuples (''prime septuplets'') and contains an overview of related sequences, e.g. the three sequences corresponding to the three admissible 8-tuples (''prime octuplets''), and the union of all 8-tuples. The first term in these sequences corresponds to the first prime in the smallest prime constellation shown below. Admissibility In order for a -tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime such that the tuple includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admissibility , ...
Admissibility may refer to: Law * Admissible evidence, evidence which may be introduced in a court of law * Admissibility (ECHR), whether a case will be considered in the European Convention on Human Rights system Mathematics and logic * Admissible decision rule, in statistical decision theory, a rule which is never dominated * Admissible rule, in logic, a type of rule of inference * Admissible heuristic, in computer science, is a heuristic which is no more than the lowest-cost path to the goal * Admissible prime k-tuple, in number theory regarding possible constellations of prime numbers * Admissible set, in mathematical logic, a transitive set satisfying the axioms of Kripke-Platek set theory * Admissible representation In mathematics, admissible representations are a well-behaved class of Group representation, representations used in the representation theory of reductive group, reductive Lie groups and locally compact group, locally compact totally disconnected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials are: : 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its primorial, , is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Hardy–Littlewood Conjecture
In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923.. Statement The conjecture states that \pi(x+y) \leq \pi(x) + \pi(y) for integers , where denotes the prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal ..., giving the number of prime numbers up to and including . Connection to the first Hardy–Littlewood conjecture The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from to is always less than or equal to the number of primes from 1 to . This was proved to be inconsistent with the firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Hardy–Littlewood Conjecture
In number theory, the first Hardy–Littlewood conjecture states the asymptotic formula for the number of prime ''k''-tuples less than a given magnitude by generalizing the prime number theorem. It was first proposed by G. H. Hardy and John Edensor Littlewood in 1923.. Statement Let m_1, m_2, \ldots, m_k be positive even integers such that the numbers of the sequence P = (p, p + m_1, p + m_2, \ldots , p + m_k) do not form a complete residue class with respect to any prime and let \pi_(n) denote the number of primes p less than n st. p + m_1, p + m_2, \ldots , p + m_k are all prime. Then :\pi_P(n)\sim C_P\int_2^n \frac, where :C_P=2^k \prod_\frac is a product over odd primes and w(q; m_1, m_2, \ldots , m_k) denotes the number of distinct residues of 0, m_1, m_2, \ldots , m_k modulo q. The case k=1 and m_1=2 is related to the twin prime conjecture. Specifically if \pi_2(n) denotes the number of twin primes less than ''n'' then :\pi_2(n)\sim C_2 \int_2^n \frac, where :C_2 = 2\pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 370,000 sequences, and is growing by approximately 30 entries per day. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. History Neil Sloane started coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A008407
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yitang Zhang
Yitang Zhang (; born February 5, 1955) is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015. Previously working at the University of New Hampshire as a lecturer, Zhang submitted a paper to the ''Annals of Mathematics'' in 2013 which established the first finite bound on the least gap between consecutive primes that is attained infinitely often. This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur Fellowship. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015. Early life and education Zhang was born in Shanghai, China, with his ancestral home in Pinghu, Zhejiang. He lived in Shanghai with his grandmother until he went to Peking University. At around the age of nine, he found a proof of the Pythagorean theorem. He first learned about Fermat's Last Theorem and Goldbach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are Infinite set, infinitely many prime number, prime numbers. It was first proven by Euclid in his work ''Euclid's Elements, Elements''. There are several proofs of the theorem. Euclid's proof Euclid offered a proof published in his work ''Elements'' (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime numbers ''p''1, ''p''2, ..., ''p''''n''. It will be shown that there exists at least one additional prime number not included in this list. Let ''P'' be the product of all the prime numbers in the list: ''P'' = ''p''1''p''2...''p''''n''. Let ''q'' = ''P'' + 1. Then ''q'' is either prime or not: *If ''q'' is prime, then there is at least one more prime that is not in the list, namely, ''q'' itself. *If ''q'' is not prime, then some prime factor ''p'' divides ''q''. If this factor ''p'' were in our list, then it wo ... [...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] |
Conjectured
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 (1.2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
