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Prime Sextuplet
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. Prime quadruplets The first eight prime quadruplets are: , , , , , , , All prime quadruplets except are of the form for some integer ''n''. (This structure is necessary to ensure that none of the four primes are divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade. A prime quadruplet can be described as a consecutive pair of twin primes, two overlapping sets of prime triplets, or two intermixed pairs of sexy primes. It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
193 (number)
193 (one hundred ndninety-three) is the natural number following 192 and preceding 194. In mathematics 193 is a prime number, and a Pierpont prime, implying that a 193-gon can be constructed using a compass, straightedge, and angle trisector. It is the only odd prime p known for which 2 is not a primitive root of 4p^2 + 1. It is the number of compositions of 14 into distinct parts. It is also part of the prime quadruplet (191, 193, 197, 199). See also * 193 (other) 193 A.D. is a year. 193 may also refer to: * 193 Ambrosia * Connecticut Route 193 * Maryland Route 193 * West Virginia Route 193 * Alabama State Route 193 * California State Route 193 * Ohio State Route 193 * Georgia State Route 193 * Maine S ... References {{DEFAULTSORT:193 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skewes's Number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which :\pi(x) > \operatorname(x), where is the prime-counting function and is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing between \pi(x) \operatorname(x) near e^ < 1.397 \times 10^. It is not known whether it is the smallest crossing. Skewes's numbers , who was Skewes's research supervisor, had proved in that there is such a number (and so, a first such number); and in ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ishango Bone
The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of Congo, is a bone tool and possible mathematical device that dates to the Upper Paleolithic era. The curved bone is dark brown in color, about 10 centimeters in length, and features a sharp piece of quartz affixed to one end, perhaps for engraving. Because the bone has been narrowed, scraped, polished, and engraved to a certain extent, it is no longer possible to determine what animal the bone belonged to, although it is assumed to belong to a mammal.Association pour la diffusion de l'information archéologique/Royal Belgian Institute of Natural Sciences, Brussels (n.d.). "Have You Heard of Ishango?" (PDF). ''Natural Sciences''. The ordered engravings have led many to speculate the meaning behind these marks, including interpretations like mathematical significance or astrological relevance. It is thought by some to be a tally stick, as it features a series of what has been interprete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cousin Prime
In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences and in OEIS) below 1000 are: :(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487, 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971) Properties The only prime belonging to two pairs of cousin primes is 7. One of the numbers will always be divisible by 3, so is the only case where all three are primes. An example of a large proven cousin prime pair is for :p = 4111286921397 \times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brun's Constant
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. Asymptotic bounds on twin primes The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let \pi_2(x) denote the number of primes ''p'' ≤ ''x'' for which ''p'' + 2 is also prime (i.e. \pi_2(x) is the number of twin primes with the smaller at most ''x''). Then, for ''x'' ≥ 3, we have : \pi_2(x) =O\left(\frac \right). That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. It follows from this bound that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms the su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Pages
The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms. , the 5,000th prime has around 412,000 digits.. Retrieved on 2018-02-12. The PrimePages has articles on primes and primality testing. It includes "The Prime Glossary" with articles on hundreds of glosses related to primes, and "Prime Curios!" with thousands of curios about specific numbers. The database started as a list of titanic primes (primes with at least 1000 decimal digits) by Samuel Yates. In subsequent years, the whole top-5,000 has consisted of gigantic primes (primes with at least 10,000 decimal digits). Primes of special forms are kept on the current lists if they are titanic and in the top-20 or top-5 for their form. See also *List of prime numbers This is a list of articles about pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime Conjecture
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sexy Prime
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes. Primorial ''n''# notation As used in this article, # stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ . Types of groupings Sexy prime pairs The sexy primes (sequences and in OEIS) below 500 are: :(5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Triplet
In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself). Examples The first prime triplets are (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887) Subpairs of primes A prime triplet contains a single pair of: *Twin primes: or ; *Cousin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
199 (number)
199 (one hundred ndninety-nine) is the natural number following 198 and preceding 200. In mathematics 199 is a centered triangular number. It is a prime number and the fourth part of a prime quadruplet: 191, 193, 197, 199. It is an emirp, given that the reversal of its digits is also prime (991). 199 is the smallest natural number that takes more than two iterations to compute its digital root as a repeated digit sum: \begin 199&\mapsto 1+9+9=19\\ &\mapsto 1+9=10\\ &\mapsto 1+0=1. \end Thus, its additive persistence is three, and it is the smallest number of persistence three. See also * The year AD 199 or 199 BC __NOTOC__ Year 199 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Lentulus and Tappulus (or, less frequently, year 555 '' Ab urbe condita''). The denomination 199 BC for this year has be ... * List of highways numbered 199 * References {{Integers, 1 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
