In mathematics, a divisibility sequence is an integer sequence math a n n in N math such that for all natural numbers m , n , math text if m mid n text then a m mid a n, math i.e., whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring mathematics ring where the concept of divisibility is defined. A strong divisibility sequence is an integer sequence math a n n in N math such that for all natural numbers m , n , math gcd a m,a n a gcd m,n . math Note that a strong divisibility sequence is immediately a divisibility sequence if math m mid n math , immediately math gcd m,n m math . Then by the strong divisibility property, math gcd a m,a n a m math and therefore math a m mid a n math . Examples Any constant sequence is a divisibility sequence. Every sequence of the form math a n kn math , for some nonzero integer k , is a divisibility sequence. Every sequence of the form math a n A n B n math for integers math A B 0 math is a divisibility sequence. The Fibonacci numbers F 0, 1, 1, 2, 3, 5, 8,... form a strong divisibility sequence. Elliptic divisibility sequence s are another class of such sequences. References cite book first1 Graham last1 Everest first2 Alf last2 van der Poorten first3 Igor last3 Shparlinski first4 Thomas last4 Ward ... journal first1 Marshall last1 Hall title Divisibility sequences of third order journal Am. J. Math ... on divisibility sequences journal Bull. Amer. Math. Soc volume 45 year 1939 pages 334&ndash 336 ... C. T. last2 Long title Divisibility properties of generalized fibonacci polynomials year 1973 page .... Math. volume 112 issue 6 year 1990 pages 985&ndash 1001 title A full characterization of divisibility ... P. Ingram author2 J. H. Silverman chapter Primitive divisors in elliptic divisibility sequences pages 243 271 External links Some http oeis.org search?q divisibility sequence divisibility sequences ... more details
refimprove date January 2011 lead too short date January 2011 A divisibility rule is a shorthand way ..., usually by examining its digits. Although there are divisibility tests for numbers in any radix , and they are all different, this article presents rules and examples only for decimal numbers. Divisibility ... smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious for others such as examining the last ... with fewer digits. Note To test divisibility by any number that can be expressed as 2 sup ... Divisor Divisibility condition Examples 1 number 1 Automatic. Any integer is divisible by 1. 2 ... page 102 text divisible by Section 3.4 Divisibility Tests , p. 102 108 ref 405 4 0 5 9 and 636 6 3 ... by the product Section 3.4 Divisibility Tests , Theorem 3.4.3, p. 107 ref 1,458 1 4 5 8 18, so it is divisible ... three digits ref name Pascal s criterion ref name last m digits 34152 Examine divisibility of just ... . Then sum the results. ref http www.tavas.net index.php?op NEArticle&sid 3358 New divisibility by 13 ... examples Divisibility by 2 First, take any even number for this example it will be 376 and note the last ... number is divisible by 2 Divisibility by 3 First, take any number for this example it will be 492 ... number 6 7 8 336 336 3 112 Divisibility by 4 The basic rule for divisibility by 4 is that if the number ... by 2, then the original number is divisible by 4 Divisibility by 5 Divisibility by 5 is easily determined ... number divided by 5 Divisibility by 6 Divisibility by 6 is determined by checking the original number to see if it is both an even number Divisibility by 2 divisible by 2 and Divisibility by 3 divisible ... for divisibility by six by taking the number 246 , dropping the last digit in the number u 24 u s 6 ... digit 0 br Tenth rightmost digit 2 br Sum 51 br 51 modulo 6 3 br Remainder 3 Divisibility by 7 ... more details
The concept of infinite divisibility arises in different ways in philosophy , physics , economics , order theory a branch of mathematics , and probability theory also a branch of mathematics . One may speak of infinite divisibility, or the lack thereof, of matter , space , time , money , or abstract mathematical objects such as the continuum theory continuum . In philosophy This theory is exposed in Plato s Timaeus dialogue dialogue Timaeus and was also supported by Aristotle . Andrew Pyle philosopher Andrew Pyle gives a lucid account of infinite divisibility in the first few pages of his Atomism and its Critics . There he shows how infinite divisibility involves the idea that there is some extended item , such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort. Many professional philosophers claim that infinite divisibility involves either a collection of an infinite number of items since there are infinite divisions, there must be an infinite collection of objects , or more rarely , point sized items , or both. Pyle states that the mathematics of infinitely divisible extensions involve neither of these that there are infinite ... . Perhaps counter intuitively, atomism is compatible with infinite divisibility. For example ... conundrum of the divisibility of matter. The multiplicity of a material object &mdash the number of its ... divisible. Infinite divisibility does not imply gap less ness the rationals do not enjoy the supremum ... . Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify. In probability distributions Main infinite divisibility probability To say that a probability ... for any finite number of intervals . This concept of infinite divisibility of probability distributions ... Molina, J.A. Rocha Arteaga, A. 2007 On the Infinite Divisibility of some Skewed Symmetric Distributions ... Nesting of Matter translation of Russian Wikipedia page DEFAULTSORT Infinite Divisibility Category Probability ... more details
In mathematics, an elliptic divisibility sequence EDS is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomial s on elliptic curve s. EDS were first defined, and their arithmetic properties studied, by Morgan Ward ref name Ward Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 1948 , 31&ndash 74. ref in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography . Definition A nondegenerate elliptic divisibility sequence EDS is a sequence of integers math var W sub n sub var sub var n var &ge 1 sub defined recursively by four initial values math var W var sub 1 sub , math var W var sub 2 sub , math var W var sub 3 sub , math var W var sub 4 sub ... , then every term math var W sub n sub var in the sequence is an integer. Divisibility property An EDS is a divisibility sequence in the sense that math m mid n Longrightarrow W m mid W n. math In particular ... term in the sequence is an integer. General recursion A fundamental property of elliptic divisibility ... to Ward. See the appendix to J. H. Silverman and N. Stephens. The sign of an elliptic divisibility ... math var D sub n sub var is also called an elliptic divisibility sequence . It is a divisibility ... ref name Einsiedler M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences ... rachthesis.ps.gz Elliptic divisibility sequences . PhD thesis, Goldsmith s College University ... Everest s EDS web page. http www.mth.uea.ac.uk h090 primeEDS.html Prime Values of Elliptic Divisibility ... of Elliptic Divisibility Sequences. Category Number theory Category Integer sequences ... more details
Unreferenced date November 2007 The base conversion divisibility test is a process that can be used to determine whether or not a certain positive natural number a can be divided evenly into a larger natural number b . It is the general case for the well known test for 9 number divisibility by nine . For other divisor s, applying this test is generally harder than figuring it out by normal division. Example Is 312 evenly divisible by 13? a 13 b 312 x a 1 14 y b base 14 184 312 in base x z 1 8 4 13 z a 13 13 1, a natural number 312 is evenly divisible by 13. Example this can be solved by another method. a 3 b 1 c 2 now, 10a 30 b 1 4c 8 thus 10a b 4c 30 1 8 39,which is divisible by 13. For 2 digit numbers if a 4b is divisible by 13 then 10a b is divisible by 13. Example is 91 divisible by 13? a 9 b 1 therefore a 4b 9 4 1 13,which is divisible by 13 thus 91 13 7 91 is divisible by 13. Dividing by nine The trick for determining if a number is divisible by nine is well known If the sum of the digits of a number is divisible by nine, then the number itself is as well. This is a special case of the general rule, made easy because no base conversion is necessary since 9 1 10, and we already use base 10. Example Is 2,340 evenly divisible by 9? a 9 b 2,340 x a 1 10 y b base 10 2,340 z 2 3 4 0 9 z a 9 9 1, a natural number 2,340 is evenly divisible by 9. Proof Any number can be expressed as math number base sum i 0 n digits i times base i math We know that under Modulo Arithmetic , math base equiv base 1 1 math Thus math number equiv base 1 sum i 0 n digits i times 1 math Category Arithmetic ... more details
In mathematics , the notion of a divisor originally arose within the context of arithmetic of whole numbers. Please see the page about divisor s for this simplest example. With the development of abstract Ring mathematics rings , of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings. Definition A nonzero element b of a commutative ring R is said to divide an element a in R notation math b mid a math if there exists an element x in R with math a b x math . We also say that b is a divisor of a , or that a is a multiple of b. Notes This definition makes sense when R is any commutative semigroup , but virtually the only time divisors are discussed is when this semi group is the multiplicative monoid of a commutative ring with identity. Also, divisors are also occasionally useful in non commutative contexts, where one must then discuss right divisor left and right divisor s. Elements a and b of a commutative ring are said to be associates if both math a mid b math and math b mid a math . The associate relationship is an equivalence relation on R , and hence divides R into disjoint sets disjoint equivalence class es each of which consists of all elements of R that are associates of any particular member of the class. Properties If R has an identity, then most statements about divisibility can be translated into statements about principal ideals. For instance, math b mid a math if and only if math a subset b math . a and b are associates if and only if math a b math u is a unit if and only if u is a divisor of every element of R u is a unit if and only if math u R math . If math a b u math where u is a unit, then a and b are associates. If R is an integral domain , then the converse is true. References cz Divisor ring theory See also Zero divisor Category Ring theory ... more details
The concepts of infinite divisibility and the Decomposable distributions decomposition of distributions arise in probability and statistics in relation to seeking families of probability distributions that might be a natural choice in certain applications, in the same way that the normal distribution is. The distributions sought correspond to random variables which are equivalent to the sums of a number of independent and identically distributed random variables , where the number of such variables can be set to any pre specified number. The term infinitely divisible characteristic function is used for the characteristic function probability theory characteristic function of any infinitely divisible distribution. ref name Lukacs Lukacs, E. 1970 Characteristic Functions , Griffin , London. p. 107 ref The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti . These distributions play a very important role in probability theory in the context of limit theorems. ref name Lukacs Lukacs, E. 1970 Characteristic Functions , Griffin , London. p. 107 ref Definition In probability theory , to say that a probability distribution F on the real line is infinitely divisible means that, for every positive integer n , there exist n statistical independence independent identically distributed random variables X sub n 1 sub , ..., X sub nn sub whose sum S sub n sub X sub n 1 sub &hellip X sub nn sub has the distribution F . Examples The Poisson distribution , the negative binomial distribution , the Gamma distribution and the degenerate ... distribution References references Dom nguez Molina, J.A. Rocha Arteaga, A. 2007 On the Infinite Divisibility ...&ndash 648 doi 10.1016 j.spl.2006.09.014 Steutel, F. W. 1979 , Infinite Divisibility in Theory and Practice ... Harn, K. 2003 , Infinite Divisibility of Probability Distributions on the Real Line Marcel Dekker . ProbDistributions Infinite divisibility Infinite divisibility in probability distributions Category ... more details
with the topic since Kant s writing. Infinite divisibility Infinite divisibility refers to the idea ... discussed the infinite divisibility of extension. Actual divisibility may be limited due to unavailability ... more details
Unreferenced date November 2008 Continuous modelling is the mathematical practice of applying a mathematical model model to continuous function continuous data data which has a potentially infinite number, and divisibility, of attributes . They often use differential equation s and are converse to discrete modelling . Modelling is generally broken down into several steps Making assumptions about the data The modeller decides what is influencing the data and what can be safely ignored. Making equations to fit the assumptions. Solving the equations. Verifying the results Various statistical tests are applied to the data and the model and compared. If the model passes the verification progress it is put into practice. External links http www.npl.co.uk scientific software research math modelling Definition by the UK National Physical Laboratory Category Applied mathematics Mathapplied stub ... more details
In mathematics , Higman s lemma states that the set of finite sequences over a finite alphabet, as partially ordered by the subsequence relation, is well quasi ordering well quasi ordered . That is, if math w 1, w 2, ldots math is an infinite sequence of words over some fixed finite alphabet, then there exist indices math i j math such that math w i math can be obtained from math w j math by deleting some possibly none symbols. More generally this remains true when the alphabet is not necessarily finite, but is itself well quasi ordered, and the subsequence relation allows the replacement of symbols by earlier symbols in the well quasi ordering of labels. This is a special case of the later Kruskal s tree theorem . References citation first Graham last Higman authorlink Graham Higman title Ordering by divisibility in abstract algebras journal Proceedings of the London Mathematical Society series 3 volume 2 issue 7 pages 326 336 year 1952 doi 10.1112 plms s3 2.1.326 Category Wellfoundedness Category Order theory Category Lemmas combin stub fr Lemme de Higman ... more details
is indecomposable. Decomposable All infinite divisibility probability infinitely divisible distributions .... Related concepts At the other extreme from indecomposability is Infinite divisibility probability infinite divisibility . Cram r s theorem shows that while the normal distribution is infinitely ... Cochran s theorem Infinite divisibility probability References Lukacs, Eugene, Characteristic Functions ... more details
Unreferenced date June 2007 Process of elimination is a method to identify an entity of interest among several ones by excluding all other entities. In education testing In educational testing , the process of elimination is a test taking tactic for increasing the chances of answering multiple choice question multiple choice questions correctly. A test taker is presented with several possibilities, of which only one answers the question. Even if only one is eliminated and the test taker guesses among the rest, it is rather more probability probable he will hit it when there are only five or four the gain in luck is substantial. Method The method of elimination is iterative algorithm iterative . One looks at the answers, determines that several answers are unfit, eliminates these, and repeats, until one cannot eliminate any more. This iteration is most effectively applied when there is logic logical structure between the answers that is to say, when by eliminating an answer one can eliminate several others. In this case one can find the answers which one cannot eliminate by eliminating any other answers and test them alone the others are eliminated as a logical consequence. This is the idea behind optimizations for computerized searches when the input is sorted as, for instance, in binary search algorithm binary search . Application Howto date January 2011 Here are two questions of one sort, to illustrate how this tactic is applied. In the first, elimination produces an answer almost at once if you know how to go at it in the other, there is no way around it you must try every answer. By which of the following is the number 2135 divisibility divisible 2, 3, 4, 15, 7? Since see divisibility rule for a refresher 2135 is not divisible by 2, it is not divisible by 4 since 2 1 3 5 11 and it is not divisible by 3, it is not divisible by 15. Then only 7 is left and, indeed 305 times 7 is 2135. Note that, if we had a number divisible by 2 but not by 4 and not divisible by 7 ... more details
and digital roots can be used for quick divisibility rule divisibility tests a natural number is divisible ... chess . Harshad number s are defined in terms of divisibility by their digit sums, and Smith ... more details
EDS may refer to TOC right Education Educational specialist Ed.S. , a terminal academic degree in the U.S. Episcopal Divinity School , an Episcopal Seminary in Cambridge, Massachusetts Evansville Day School , an independent college prep school in Evansville, Indiana Politics Environmental Defence Society , a New Zealand environmental organisation European Democrat Students , a centre right political students union Evropsk demokratick strana , a Czech political party Science and mathematics Electrodynamic suspension , a type of magnetic levitation Elliptic divisibility sequence , a class of integer sequences in mathematics Energy dispersive X ray spectroscopy , a method used to determine the energy spectrum of X ray radiation Medicine Ehlers Danlos syndrome , a group of heritable connective tissue disorders Episodic dyscontrol syndrome , a pattern of episodic, abnormal, and often violent and uncontrollable social behavior Excessive daytime sleepiness , a sleep disorder symptom, especially common in Sleep Apnea and Narcolepsy Technology Earth Departure Stage , the second stage of the Ares V and Block II Space Launch System launch vehicles Electronic Document System , an early graphical hypertext system Electronic Datasheet, a standard for field device description in CANopen based automation systems Explosive detection system, a mechanism for detecting explosive material Extended Data Services now XDS , a standard for the delivery of metadata on NTSC video signals Business Electronic Data Systems , a technology company founded by Ross Perot, now HP Enterprise Services disambiguation cs EDS de EDS fr EDS it EDS lv EDS nl EDS ja EDS pl EDS ujednoznacznienie sk EDS sl EDS zh EDS ... more details
by 101. This might not be as simple as the divisibility tests for numbers like 3 or 5, and it might not be terribly practical, but it is simpler than the divisibility tests for other 3 digit ... more details
Unreferenced date December 2009 Cleanup date May 2010 In commodity money , intrinsic value can be partially or entirely due to the desirable features of the object as a medium of exchange and a store of value . Examples of such features include divisibility easily and securely storable and transportable scarcity and hard to counterfeit. When objects come to be used as a medium of exchange they lower the high transaction costs associated with barter and other in kind transactions. In numismatics , intrinsic value is the value of the metal, typically a precious metal, in a coin . For example, if gold trades in commercial markets at a price of Federal money this effect can, at the margin, mitigate forces that are known to cause inflation . When copper prices skyrocketed due to over issuance of Federal Reserve Notes in the mid to late 1970s, there was a fear that the Penny U.S. coin U.S. one cent piece might succumb to this fate. In fact, this did happen, leading the United States Mint Mint to change the composition of the cent in 1982 to allow convertibility between the two competing currencies Federal Reserve Notes issued for profit by a private corporation and United States coins Pursuant to Title 31 Section 5111 of the US Code and under the authority of the Coinage Act of 1792 and the constitution. class wikitable style margin 1em auto 1em auto text align left Intrinsic Value The market value of the constituent metal within a coin. Legal or Face Value The legally defined value of a coin relative to other units of currency. Market Value The price that a coin will fetch in the marketplace. For most coins in circulation this value is coincident with the face value. See also Marginal theory of value Labor theory of value DEFAULTSORT Intrinsic Value Numismatics Category Numismatics nl Intrinsieke waarde ... more details
William Heytesbury ref Known as Gugliemus Hentisberus or Tisberus. ref ca. 1313 1372 1373 , philosopher and logician, is best known as one of the Oxford Calculators of Merton College , where he was a fellow by 1330 . In his work he applied logical techniques to the problems of Infinitely divisible divisibility , the Continuum set theory continuum , and kinematics . His masterpiece magnum opus was the Regulae solvendi sophismata Rules for Solving Sophism s , written c. 1335 . He was Chancellor education Chancellor of the University of Oxford from 1371 until 1372. Works 1335 Regulae solvendi sophismata Rules for Solving Sophisms 1. On insoluble sentences 2. On knowing and doubting 3. On relative terms 4. On beginning and ceasing 5. On maxima and minima 6. On the three categories 1483 De probationibus conclusionum tractatus regularum solvendi sophismata , Pavia 1483 De tribus praedicamentis De probationibus conclusionum tractatus regularum solvendi sophismata On the Proofs of Conclusions from the Treatise of Rules for Resolving Syllogisms Liber Calculationum Further reading Sylla, Edith 1982 The Oxford Calculators , in Norman Kretzmann , Anthony Kenny & Pinborg edd. , The Cambridge History of Later Medieval Philosophy Murdoch, John 1982 Infinity and Continuity , in Kretzmann, Kenny & Pinborg edd. , The Cambridge History of Later Medieval Philosophy References sep entry heytesbury William Heytesbury John Longeway Notes references DEFAULTSORT Heytesbury, William Category 14th century mathematicians Category 14th century philosophers Category 14th century English people Category 14th century Latin writers Category Scholastic philosophers Category Fellows of Merton College, Oxford Category Medieval European mathematics philosopher stub de William Heytesbury fr William Heytesbury nl William van Heytesbury pt William de Heytesbury ru , ... more details
In mathematics , a Wieferich pair is a pair of prime number s p and q that satisfy p sup q &minus 1 sup 1 Modular arithmetic Congruence relation mod q sup 2 sup and q sup p &minus 1 sup 1 mod p sup 2 sup Wieferich pairs are named after Germany German mathematician Arthur Wieferich . Wieferich pairs play an important role in Preda Mih ilescu s 2002 proof ref cite journal author Preda Mih ilescu authorlink Preda Mih ilescu title Primary Cyclotomic Units and a Proof of Catalan s Conjecture journal J. Reine Angew. Math. volume 572 year 2004 pages 167 195 mr 2076124 ref of Mih ilescu s theorem formerly known as Catalan s conjecture . ref Jeanine Daems http www.math.leidenuniv.nl jdaems scriptie Catalan.pdf A Cyclotomic Proof of Catalan s Conjecture . ref Known Wieferich pairs There are only six Wieferich pairs known ref MathWorld title Double Wieferich Prime Pair urlname DoubleWieferichPrimePair ref 2, 1093 , 3, 1006003 , 5, 1645333507 , 83, 4871 , 911, 318917 , and 2903, 18787 sequences OEIS2C id A124121 and OEIS2C id A124122 in On Line Encyclopedia of Integer Sequences OEIS See also Wieferich prime Fermat quotient References Reflist Further reading cite journal author Yuri Bilu title Catalan s conjecture after Mih ilescu journal Ast risque volume 294 year 2004 pages vii, 1&ndash 26 cite journal author R. Ernvall coauthors T. Mets nkyl title On the p divisibility of Fermat quotients journal Math. Comp. volume 66 issue 219 year 1997 pages 1353 1365 url http www.ams.org mcom 1997 66 219 S0025 5718 97 00843 0 home.html doi 10.1090 S0025 5718 97 00843 0 cite journal author Ray Steiner title Class number bounds and Catalan s equation journal Math. Comp. volume 67 issue 213 year 1998 pages 1317 1322 url http www.ams.org mcom 1998 67 223 S0025 5718 98 00966 1 home.html doi 10.1090 S0025 5718 98 00966 1 Category Prime numbers ... more details
terms this gives type A sub 3 sub , another divisibility condition giving type A sub 4 sub , and a final non divisibility condition giving type exactly A sub 4 sub . To see where these extra divisibility ... quartic order four in x sub 1 sub and y sub 1 sub . The divisibility condition for type A sub ... more details
In mathematics, Atkin Lehner theory is part of the theory of modular form s, in which the concept of newform is defined in such a way that the theory of Hecke operators can be extended to higher level. A newform is a cusp form new at a given level N , where the levels are the nested subgroups &Gamma sub 0 sub N of the modular group , with N ordered by divisibility . That is, if M divides N , sub 0 sub N is a subgroup of sub 0 sub M . The oldforms for sub 0 sub N are those modular forms f &tau of level N of the form g d &tau for modular forms g of level M with M a proper divisor of N , where d divides N M . The newforms are defined as a vector subspace of the modular forms of level N , complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product . The Hecke operator s, which act on the space of all cusp forms, preserve the subspace of newforms and are self adjoint and commuting operators with respect to the Petersson inner product when restricted to this subspace. Therefore, the algebra of operators on newforms they generate is a finite dimensional C algebra that is commutative and by the spectral theory of such operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra . References Citation authorlink A. O. L. Atkin last1 Atkin first1 A. O. L. authorlink2 Joseph Lehner last2 Lehner first2 J. title Hecke operators on sub 0 sub m doi 10.1007 BF01359701 mr 0268123 year 1970 journal Mathematische Annalen issn 0025 5831 volume 185 pages 134 160 Category Modular forms ... more details
Orphan date June 2011 The rule of nines , in mathematics, is a divisibility rule for the divisor 9. It is notable because it illustrates some interesting properties of modular arithmetic , and its proof is derived from that basis. The rule is that any positive integer is divisible by 9 if and only if the sum of its digits is also divisible by 9, when expressed in decimal decimal notation . Proof This proof, although not directly taken from that source, is based on the one by Flannery 2001 . Let the positive integer n be represented by the decimal digits a sub k sub a sub k 1 sub ... a sub 2 sub a sub 1 sub a sub 0 sub . Because math begin align 10 0 & equiv 1 pmod 9 10 1 & equiv 1 pmod 9 10 2 & equiv 1 pmod 9 &... end align math and multiplication functions the same way in modular arithmetic as it does in elementary algebra , ref with the caveat that the modulus must remain the same ref math begin align a 0 times 10 0 & equiv a 0 pmod 9 a 1 times 10 1 & equiv a 1 pmod 9 a 2 times 10 2 & equiv a 2 pmod 9 &... a k times 10 k & equiv a k pmod 9 . end align math Summing these equivalences, we get math a k times 10 k ... a 0 times 10 0 equiv a k ... a 0 pmod 9 . math Notice that the left term of this equivalence is equal to n , according to our definition. Therefore, the sum of the digits of n is equivalent to n itself modulo 9 and so this sum is divisible by 9 if and only if n is also divisible by 9. The proof is complete. Notes Not for actual references, it s for the note in the ref tag above references References cite book last Flannery title In Code A Mathematical Journey publisher Workman Publishing year 2001 isbn 0761123849 Category Elementary arithmetic Category Articles containing proofs ... more details
An instant is a infinitesimal moment in time , a moment whose passage is instantaneous. The continuous nature of time and its infinite divisibility was addressed by Aristotle in his Physics Aristotle Physics where he wrote on Zeno s paradoxes . The philosopher and mathematician Bertrand Russell was still seeking to define the exact nature of an instant thousands of years later. ref citation url http books.google.co.uk books?id 29E9AAAAIAAJ&pg PA129 title The structure of time author W. Newton Smith chapter The Russellian construction of instants page 129 publisher Routledge year 1984 isbn 9780710203892 ref In physics , a theoretical lower bound unit of time called the Planck time has been proposed, that being the time required for light to travel a distance of 1 Planck length . ref name gsu hbase cite web url http hyperphysics.phy astr.gsu.edu hbase astro planck.html title Big Bang models back to Planck time publisher Georgia State University date 19 June 2005 ref The Planck time is theorized to be the smallest time measurement that will ever be possible, ref cite encyclopedia url http astronomy.swin.edu.au cosmos P Planck Time title Planck Time encyclopedia COSMOS The SAO Encyclopedia of Astronomy publisher Swinburne University ref roughly 10 sup 43 sup seconds. Within the framework of the laws of physics as we understand them today, for times less than one Planck time apart, we can neither measure nor detect any change. As of May 2010, the smallest time interval that was directly measured was on the order of 12 attoseconds 12 10 sup 18 sup seconds , ref cite web url http www.physorg.com news192909576.html title 12 attoseconds is the world record for shortest controllable time ref about 10 sup 24 sup times larger than the Planck time. It is therefore physically impossible, with current technology, to determine if any action exists that causes a reaction in an instant , rather than a reaction occurring after an interval of time too short to observe or measure. See a ... more details
for the related pairing on the Tate Shafarevich group Cassels Tate pairing In mathematics, Tate pairing is any of of several closely related bilinear pairing s involving elliptic curve s or abelian varieties , usually over local field local or finite field s, based on the Tate duality pairings introduced by harvs txt last Tate authorlink John Tate year1 1958 year2 1963 and extended by harvtxt Lichtenbaum 1969 . harvtxt R ck Frey 1995 applied the Tate pairing over finite fields to cryptography. References Citation last1 Lichtenbaum first1 Stephen title Duality theorems for curves over p adic fields doi 10.1007 BF01389795 id MR 0242831 year 1969 journal Inventiones Mathematicae issn 0020 9910 volume 7 pages 120 136 Citation last1 R ck first1 Hans Georg last2 Frey first2 Gerhard title A remark concerning m divisibility and the discrete logarithm in the divisor class group of curves url http dx.doi.org 10.2307 2153546 doi 10.2307 2153546 id MR 1218343 year 1994 journal Mathematics of Computation issn 0025 5718 volume 62 issue 206 pages 865 874 Citation last1 Tate first1 John author1 link John Tate title WC groups over p adic fields url http www.numdam.org item?id SB 1956 1958 4 265 0 publisher Secr tariat Math matique location Paris series S minaire Bourbaki 10e ann e 1957 1958 id MR 0105420 year 1958 volume 13 Citation last1 Tate first1 John author1 link John Tate title Proceedings of the International Congress of Mathematicians Stockholm, 1962 url http mathunion.org ICM ICM1962.1 publisher Inst. Mittag Leffler location Djursholm id MR 0175892 year 1963 chapter Duality theorems in Galois cohomology over number fields pages 288 295 Category Pairing based cryptography Category Elliptic curve cryptography Category Elliptic curves ... more details
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