In abstract algebra , a principal ideal domain , or PID , is an integral domain in which every ideal ring theory ideal is principal ideal principal , i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors e.g., Bourbaki refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisor s whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects which behave somewhat like the integers , with respect to Integral domain Divisibility, prime and irreducible elements divisibility any element of a PID has a unique decomposition into Integral domain Divisibility, prime and irreducible elements prime elements so an analogue of the fundamental theorem of arithmetic holds any two elements of a PID have a greatest common divisor although it may not be possible to find it using the Euclidean algorithm . If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax by . Principal ideal domains are noetherian ring noetherian , they are integrality integrally closed , they are unique factorization domain s and Dedekind domain Dedekind rings . All Euclidean domain s and all field mathematics fields are principal ideal domains. Commutative ring s integral domain s integrally closed domain s unique factorization domain s principal ideal domains Euclidean domain s field mathematics field s Examples Examples include K any field mathematics field , Z the ring mathematics ring of integer s, ref See Fraleigh & Katz 1967 , p. 73, Corollary of Theorem 1.7, and notes at p. 369, after the corollary of Theorem 7.2 ref K x polynomial ring rings of polynomials in one variable with coefficients in a field. The converse is also true that is, if A x is a PID, then A is a field. Furthermore, a ring of formal power series in one variable over a field is a P ... more details
v onepage&q commercial 20chamber 20arbitration 20Menyh C3 A1rd&f false title Aggregation and divisibility ... Law and Civil Law Perspectives. Wien New York Springer Verlag, 2009 . Aggregation and Divisibility of Damage in Hungary Tort Law and Insurance ., Aggregation and Divisibility of Damage. Wien New York ... more details
the following axioms to the axiomatic system of monoidal t norm logic Divisibility math A wedge B rightarrow ... logic BL , or by adding the axiom of divisibility to the logic IMTL. Real valued semantics Infinite ... more details
Refimprove date January 2010 Textbook date January 2010 In mathematics , a binary relation R on a Set mathematics set X is antisymmetric if, for all a and b in X if R a,b and R b,a , then a b , or, equivalently, if R a,b with a b , then R b,a must not hold. In mathematical notation , this is math forall a, b in X, R a,b and R b,a Rightarrow a b math or, equivalently, this is the same formula as above, but due to the addition of the negation, it is more clear where the term anti symmetric comes from math forall a, b in X, R a,b and a ne b Rightarrow lnot R b,a . math The usual order relation on the real number s is antisymmetric if for two real numbers x and y both inequality mathematics inequalities x y and y x hold then x and y must be equal. Similarly, the subset order on the subsets of any given set is antisymmetric given two sets A and B , if every Element mathematics element in A also is in B and every element in B is also in A , then A and B must contain all the same elements and therefore be equal math A subseteq B and B subseteq A Rightarrow A B math partial order Partial and total order s are antisymmetric by definition. A relation can be both symmetric relation symmetric and antisymmetric e.g., equality mathematics the equality relation , and there are relations which are neither symmetric nor antisymmetric e.g., the preys on relation on biological species . Antisymmetry is different from Asymmetric relation asymmetry . According to one definition of asymmetric , anything that fails to be symmetric is asymmetric. Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus reflexive relation irreflexivity . Examples The relation x is even, y is odd between a pair x , y of integer s is antisymmetric Image Evenandodd.PNG The divisibility order of the natural number s is another example of an antisymmetric relation. See also Symmetry in mathematics References MathWorld urlname AntisymmetricRe ... more details
In abstract algebra , an element math p math of a commutative ring math R math is said to be prime if it is not zero, not a unit ring theory unit and whenever math p math Divisibility ring theory divides math ab math for some math a math and math b math in math R math , then math p math divides math a math or math p math divides math b math . Equivalently, an element math p math is prime if, and only if, the principal ideal math p math generated by math p math is a nonzero prime ideal . ref harvnb Hungerford 1980 loc Theorem III.3.4 i , as indicated in the remark below the theorem and the proof, the result holds in full generality. ref Interest in prime elements comes from the Fundamental theorem of arithmetic , which asserts that each integer can be written in essentially only one way as 1 or 1 multiplied by a product of positive prime number s. This led to the study of unique factorization domain s, which generalize what was just illustrated in the integers. Prime elements should not be confused with irreducible element s. In an integral domain , every prime is irreducible ref harvnb Hungerford 1980 loc Theorem III.3.4 iii ref but the converse is not true in general. However, in unique factorization domains, ref harvnb Hungerford 1980 loc Remark after Definition III.3.5 ref or more generally in GCD domain s, primes and irreducibles are the same. Being prime is also relative to which ring an element is considered to be in for example, 2 is a prime element in Z but it is not in Z math i math , the ring of Gaussian integers , since math 2 i 1 i 2 math and 2 does not divide any factor on the right. Examples The following are examples of prime elements in rings The integers 2, 3, 5, 7, 11,... in the ring of integers Z the complex numbers math 1 i math , 19, and math 2 3i math in the ring of Gaussian integers Z math i math the polynomials math x 2 2 math and math x 2 1 math in the ring of polynomials over Z . References Notes reflist Sources Section III.3 of Citation au ... more details
A palindromic prime sometimes called a palprime is a prime number that is also a palindromic number . Palindromicity depends on the Radix base of the numbering system and its writing conventions, while primality is independent of such concerns. The first few decimal palindromic primes are 2 number 2 , 3 number 3 , 5 number 5 , 7 number 7 , 11 number 11 , 101 number 101 , 131 number 131 , 151 number 151 , 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, OEIS id A002385 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. The largest known As of 2011 11 lc on is 10 sup 205030 sup 7047407 10 sup 102512 sup 1, found by Harvey Dubner. ref Chris Caldwell, http primes.utm.edu top20 page.php?id 53 The Top Twenty Palindrome ref On the other hand, it is known that, for any base, almost all palindromic numbers are composite. ref http www.esi.ac.at preprints esi1456.pdf Banks et al. ref In binary numeral system binary , the palindromic primes include the Mersenne prime s and the Fermat prime s. All binary palindromic primes except binary 11 decimal 3 have an odd number of digits those palindromes with an even number of digits are divisible by 3. The sequence of binary palindromic primes begins in binary 11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, OEIS id A117697 Ribenboim defines a triply palindromic prime as a prime p for which p is a palindromic prime with q digits, where q is a palindromic prime with r digits, where r is also a palindromic prime. ref Paulo Ribenboim , The New Book of Prime Number Records ref For example, p 10 sup 11310 sup 4661664 e 5652 1, which has q 11311 digits, and 11311 has r 5 digits. The first base 10 triply palindromic prime is the 11 digit 10000500001. It s ... more details
In mathematics , the supernatural numbers sometimes called generalized natural numbers or Steinitz numbers are a generalization of the natural number s. They were used by Ernst Steinitz ref Ernst Steinitz, http resolver.sub.uni goettingen.de purl?GDZPPN002167042 Algebraische Theorie der K rper , Journal f r die reine und angewandte Mathematik 1910 , pp. 167 309. ref in 1910 as a part of his work on Field theory mathematics field theory . A supernatural number math omega math is a formal Product mathematics product math omega prod p p n p , math where math p math runs over all prime number s, and each math n p math is either a zero, natural number or infinity . Sometimes we write math v p omega math for math n p math . If no math n p infty math and there are only a finite number of non zero math n p math then we recover the natural numbers. Slightly less intuitively, if all math n p math are math infty math , we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide math omega math infinitely often, by taking that prime s corresponding exponent to be the symbol math infty math . There is no natural way to add supernatural numbers, but they can be multiplied, with math prod p p n p cdot prod p p m p prod p p n p m p math . Similarly, the notion of divisibility extends to the supernaturals with math omega 1 mid omega 2 math if math v p omega 1 leq v p omega 2 math for all math p math . We can also generalize the notion of the least common multiple and greatest common divisor for supernatural numbers, by defining math displaystyle operatorname lcm omega i displaystyle prod p p sup v p omega i math math displaystyle operatorname gcd omega i displaystyle prod p p inf v p omega i math With these definitions, we can now take the gcd or lcm of infinitely many natural numbers to get a supernatural number. We can also extend the usual p adic number math p math adic ord ... more details
In abstract algebra , a subset of a ring mathematics ring is said to be multiplicatively closed if it is closure mathematics closed under multiplication i.e., xy is in the set when x and y are in it and contains 1 but doesn t contain 0. ref Zero is excluded since the multiplicative sets useful for localization exclude zero, and because a saturated set containing 0 is the whole ring, another undesirable triviality. ref The condition is especially important in commutative algebra , where multiplicatively closed sets are used to build localization of a ring localizations of commutative rings. Examples Common examples of multiplicatively closed sets include the set math 1, x, x 2, x 3, dots math where x is not a nilpotent element the set of unit ring theory unit s of the ring the set of zero divisor regular element s of the ring in a commutative ring, the set theoretic complement of a prime ideal . Properties For commutative rings, the complement of a prime ideal is an especially important example of a multiplicatively closed set. Clearly an ideal A of a commutative ring R is prime if and only if the complement R A is multiplicatively closed. In fact, complements of prime ideals enjoy another property that of being saturated . A set is said to be saturated if every divisibility ring theory divisor of x in the set is also in the set i.e., if xy is in the set, then x and y are in the set . For a commutative ring the converse is not always true a saturated multiplicative set may not be a complement of a prime ideal. However it is true that a subset S is saturated and multiplicatively closed if and only if S is the set theoretic complement of a non empty set theoretic union of prime ideals, harv Kaplansky 1974 loc Theorem 2 p 2 . The intersection of a family of multiplicative sets is again multiplicative, and the intersection of a family of saturated sets is saturated. Suppose S is a multiplicatively closed subset of a commutative ring R . A standard lemma due to Krull ref ... more details
In number theory , Sophie Germain s theorem is a statement about the divisibility of solutions to the equation x sup p sup y sup p sup z sup p sup of Fermat s Last Theorem . Specifically, Sophie Germain proved that the product xyz must be divisible by p sup 2 sup if an auxiliary prime can be found such that two conditions are satisfied No two p sup th sup powers differ by one modular arithmetic modulo and p is itself not a p sup th sup power modular arithmetic modulo . Conversely, the first case of Fermat s Last Theorem the case in which p does not divide xyz must hold for every prime p for which even one auxiliary prime can be found. Germain identified such an auxiliary prime for every prime less than 100. The theorem and its application to primes p less than 100 were attributed to Germain by Adrien Marie Legendre in 1823. ref cite journal author Adrien Marie Legendre Legendre AM year 1823 title Recherches sur quelques objets d analyse ind termin e et particuli rement sur le th or me de Fermat journal M m. Acad. Roy. des Sciences de l Institut de France volume 6 Didot, Paris, 1827. Also appeared as Second Suppl ment 1825 to Essai sur la th orie des nombres , 2nd edn., Paris, 1808 also reprinted in Sphinx Oedipe 4 1909 , 97 128. ref Notes references References Laubenbacher R, Pengelley D 2007 http www.math.nmsu.edu 7Edavidp germain.pdf Voici ce que j ai trouv Sophie Germain s grand plan to prove Fermat s Last Theorem cite book author Louis Mordell Mordell LJ year 1921 title Three Lectures on Fermat s Last Theorem publisher Cambridge University Press location Cambridge pages 27 31 cite book author Paulo Ribenboim Ribenboim P year 1979 title 13 Lectures on Fermat s Last Theorem publisher Springer Verlag location New York isbn 978 0387904320 pages 54 63 Category Theorems in number theory fr Th or me de Sophie Germain ko ... more details
In mathematics, more specifically general topology , the divisor topology is an example of a topology given to the set X of positive integer s that are greater than or equal to two, i.e., nowrap 1 X 2, 3, 4, 5, &hellip . The divisor topology is the poset topology for the partial order relation of divisibility on the positive integers. To give the set X a topology means to say which subset s of X are open , and to do so in a way that the following axiom s are met ref name CEIT Citation first L. A. last Steen first2 J. A. last2 Seebach title Counterexamples in Topology publisher Dover year 1995 ISBN 048668735X ref The union mathematics union of open sets is an open set. The finite intersection mathematics intersection of open sets is an open set. The set X and the empty set are open sets. Construction The set X and the empty set are required to be open sets, and so we define X and to be open sets in this topology. Denote by Z sup sup the set of positive integer s, i.e., the set of positive whole number greater than or equal to one. Read the notation x n as x divides n , and consider the sets math S n x in bold Z x n math Then the set S sub n sub is the set of divisor s of n . For different values of n , the sets S sub n sub are used as a basis topology basis for the divisor topology. ref name CEIT The open sets in this topology are the lower set s for the partial order defined by nowrap 1 x y if x &thinsp &thinsp y . Properties The set of prime number s is dense topology dense in X . In fact, every dense open set must include every prime, and therefore X is a Baire space . ref name CEIT X is a Kolmogorov space that is not T1 space T1 . In particular, it is Hausdorff space non Hausdorff . X is second countable space second countable . X is connected space connected and locally connected . X is not compact space compact , since the basic open sets S sub n sub comprise an infinite covering with no finite subcovering. X is not locally compact . The closure of a point in x ... more details
Morgan Ward 1901 1963 was an American mathematician, a professor of mathematics at the California Institute of Technology . ref citation title Prof. Ward of Caltech Dies at 61 journal Los Angeles Times date June 27, 1963 . ref Ward received his Ph.D. from Caltech in 1928, with a dissertation entitled The Foundations of General Arithmetic his advisor was Eric Temple Bell . ref name mg mathgenealogy id 10385 name Morgan Ward . ref He became a research fellow at Caltech, and then in 1929 a member of the faculty he remained at Caltech until his death in 1963. ref name wardcol http www.archives.caltech.edu search catalog.cfm?results file Detail View&recsPerPage 1&firstRecToShow 2&search field Morgan 20Ward&entry type NonPhoto&photo id &cat series Collection Profile Morgan Ward 1901 1963 , Caltech Library, retrieved 2010 09 12. ref Among his doctoral students was Robert P. Dilworth , who also became a Caltech professor. ref name mg Ward is the academic genealogy academic ancestor of over 500 mathematicians and computer scientists through Dilworth and another of his students, Donald Darling. ref name mg Ward s research interests included the study of recurrence relation s and the divisibility properties of their solutions, diophantine equation s including Euler s sum of powers conjecture and equations between monomial s, abstract algebra , lattice theory and residuated lattice s, functional equation s and Iterated function functional iteration , and numerical analysis . ref name l93 citation first D. H. last Lehmer authorlink Derrick Henry Lehmer title The mathematical work of Morgan Ward journal Mathematics of Computation volume 61 issue 203 pages 307 311 doi 10.1090 S0025 5718 1993 1182245 3 . ref He also worked with the National Science Foundation on the reform of the elementary school mathematics curriculum, ref name wardcol and with Clarence Ethel Hardgrove he wrote the textbook Modern Elementary Mathematics Addison Wesley, 1962 . Ward s works are collected in the Cal ... more details
Presburger arithmetic is the first order predicate calculus first order theory of the natural number s with addition , named in honor of Moj esz Presburger , who introduced it in 1929. The signature mathematical logic signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The axioms include a schema of induction. Presburger arithmetic is much weaker than Peano arithmetic , which includes both addition and multiplication operations. Unlike Peano arithmetic, Presburger arithmetic is a Decidability logic decidable theory . This means it is possible to effectively determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic. The asymptotic running time computational complexity of this decision problem is doubly exponential, however, as shown by Fischer and Rabin 1974 . Overview The language of Presburger arithmetic contains constants 0 and 1 and a binary function , interpreted as addition. In this language, the axioms of Presburger arithmetic are the universal closure s of the following 0 x 1 x 1 y 1 x y x 0 x x y 1 x y 1 Let P x be a first order logic first order formula in the language of Presburger arithmetic with a free variable x and possibly other free variables . Then the following formula is an axiom P 0 &and &forall x P x &rarr P x 1 &rarr &forall y P y . 5 is an axiom schema of Mathematical Induction induction , representing infinitely many axioms. Since the axioms in the schema in 5 cannot be replaced by any finite number of axioms, Presburger arithmetic is not finitely axiomatizable. Presburger arithmetic cannot formalize concepts such as divisibility or prime number . Generally, any number concept leading to multiplication cannot be defined in Presburger arithmetic, since that leads to incompleteness and undecidability. However, it can formulate individual instances of divisibility for exampl ... more details
. Examples Positive integers ordered by divisibility The M bius function is a, b b a , where ... the positive integers ordered by divisibility by a positive integer corresponding to its multiset of prime ... more details
Heavy tailed distribution Truncated distribution Infinite divisibility Stability probability ... Lyapunov s central limit theorem Martingale central limit theorem Infinite divisibility probability ... more details
. 5, March April 1993. ref Fractured Paintings Image Indefinite Divisibility 1996.jpg right thumb Indefinite Divisibility , 63x51 1996 The images are based on old photographs from Russia which Edmonson ... more details
The famous problem in number theory known as Fermat s Last Theorem has repeatedly received attention in fiction and popular culture. In the Doctor Who episode The Eleventh Hour Doctor Who The Eleventh Hour , the Doctor provides the real proof of Fermat s theorem presumably Fermat s Last Theorem to show his bona fides . In The Royale TNG episode The Royale , an episode first aired 27 March 1989 of Star Trek The Next Generation , Jean Luc Picard Captain Picard states that the theorem had gone unsolved for 800 years. At the end of the episode Jean Luc Picard Captain Picard says, Like Fermat s theorem, it is a puzzle we may never solve. Wiles proof was released five years after the particular episode aired. This was subsequently mentioned in a Star Trek Deep Space Nine episode called Facets DS9 episode Facets during June 1995 in which Jadzia Dax comments that one of her previous hosts, Tobin Dax Tobin Dax Tobin Dax , had the most original approach to the proof since Wiles over 300 years ago. This reference was generally understood by fans to be a retroactive continuity for The Royale . A sum, proved impossible by the theorem, appears in an episode of The Simpsons , Treehouse of Horror VI . In the three dimensional world in Homer sup 3 sup , the equation math 1782 12 1841 12 1922 12 math is visible, just as the dimension begins to collapse. The joke is that the twelfth root of the sum does evaluate to 1922 due to rounding errors when entered into most handheld calculators notice that the left hand side is odd, while math 1922 12 math is even, so the equality cannot hold. Instead of 1922, it actually is 1921.999999995. A second counterexample appeared in a later episode, The Wizard of Evergreen Terrace math 3987 12 4365 12 4472 12 math . These agree to 10 of 44 decimal digits, but notice that simple Divisibility rule 2 through 20 divisibility rules show 3987 and 4365 are multiples of 9 so that a sum of their powers is also. The same rule reveals that 4472 is not divisible ... more details
In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties it is the largest natural number N such that the equation Nx g has a solution x &isin A , or symbol &infin if the largest number with this property does not exist. The p height considers only divisibility properties by the powers of a fixed prime number p . The notion of height admits a refinement so that the p height becomes an ordinal number . Height plays an important role in Pr fer theorems and also in Ulm s theorem , which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants . Definition of height Let A be an abelian group and g an element of A . The p height of g in A , denoted h sub p sub g , is the largest natural number n such that the equation p sup n sup x g has a solution in x &isin A , or the symbol &infin if a solution exists for all n . Thus h sub p sub g n if and only if g &isin p sup n sup A and g ¬in p sup n 1 sup A . This allows one to refine the notion of height. For any ordinal &alpha , there is a subgroup p sup sup A of A which is the image of the multiplication map by p iterated times, defined using transfinite induction p sup 0 sup A A p sup 1 sup A p p sup &alpha sup A p sup sup A &cap sub sub p sup sup A if is a limit ordinal . The subgroups p sup sup A form a decreasing filtration of the group A , and their intersection is the subgroup of the p divisible elements of A , whose elements are assigned height &infin . The modified p height h sub p sub sup &lowast sup g if g &isin p sup sup A , but g ¬in p sup 1 sup A . The construction of p sup sup A is functorial in A in particular, subquotients of the filtration are isomorphism invariants of A . Ulm subgroups Let p be a fixed prime number. The first Ulm subgroup of an abelian group A , denoted U A or A sup 1 sup , is p sup sup A &cap sub n sub p ... more details
number s equipped with the relation of divisor Divisibility of numbers divisibility . The vertex ... element is a lower bound of P . For example, consider the positive integer s, ordered by divisibility ... exclude the number 1, while keeping divisibility as ordering on the elements greater than 1, then the resulting ... more details
In number theory , a Woodall number W sub n sub is any natural number of the form W sub n sub n 2 sup n sup 1 for some natural number n . The first few Woodall numbers are 1, 7, 23, 63, 159, 383, 895, OEIS id A003261 . Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917 Citation needed December 2011 date December 2011 , inspired by James Cullen mathematician James Cullen s earlier study of the similarly defined Cullen number s. Woodall numbers curiously arise in Goodstein s theorem . Citation needed date October 2009 Woodall numbers that are also prime number s are called Woodall primes the first few exponents n for which the corresponding Woodall numbers W sub n sub are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, OEIS id A002234 the Woodall primes themselves begin with 7, 23, 383, 32212254719, OEIS id A050918 . In 1976 Christopher Hooley showed that almost all Cullen numbers are composite number composite Citation needed December 2011 date December 2011 . Hooley s proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n 2 sup n a sup b where a and b are integers Citation needed December 2011 date December 2011 , and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes Citation needed December 2011 date December 2011 . As of 2007 12 , the largest known Woodall prime is 3752948 2 sup 3752948 sup 1. ref Citation url http primes.utm.edu primes page.php?id 83407 title The Prime Database 938237 2 3752950 1 work Chris Caldwell s The Largest Known Primes Database accessdate December 22, 2009 ref It has 1,129,757 digits and was found by Matthew J. Thompson in 2007 in the distributed computing project PrimeGrid . Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides W sub p 1 2 sub if the Jacobi symbol math left frac 2 p right math ... more details
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